Using modeling to determine ion energy associated with a plasma system

ABSTRACT

Systems and methods for determining ion energy are described. One of the methods includes detecting output of a generator to identify a generator output complex voltage and current (V&amp;I). The generator is coupled to an impedance matching circuit and the impedance matching circuit is coupled to an electrostatic chuck (ESC). The method further includes determining from the generator output complex V&amp;I a projected complex V&amp;I at a point along a path between an output of a model of the impedance matching circuit and a model of the ESC. The operation of determining of the projected complex V&amp;I is performed using a model for at least part of the path. The method includes applying the projected complex V&amp;I as an input to a function to map the projected complex V&amp;I to a wafer bias value at the ESC model and determining an ion energy from the wafer bias value.

CLAIM OF PRIORITY

The present patent application is a continuation of and claims thebenefit of and priority, under 35 U.S.C. § 120, to U.S. patentapplication Ser. No. 14/184,639, filed on Feb. 19, 2014, and titled“Using Modeling To Determine Ion Energy Within A Plasma System”, whichis incorporated by reference herein in its entirety.

The application Ser. No. 14/184,639 claims the benefit of and priority,under 35 U.S.C. § 119(e), to U.S. Provisional Patent Application No.61/799,969, filed on Mar. 15, 2013, and titled “Using Modeling ToDetermine Ion Energy Within A Plasma System”, now issued as U.S. Pat.No. 9,502,216, which is hereby incorporated by reference in itsentirety.

The application Ser. No. 14/184,639 is a continuation-in-part of andclaims the benefit of and priority, under 35 U.S.C. § 120, to U.S.patent application Ser. No. 13/756,390, filed on Jan. 31, 2013, andtitled “Using Modeling to Determine Wafer Bias Associated with a PlasmaSystem”, which is incorporated by reference herein in its entirety.

FIELD

The present embodiments relate to using modeling to determine ion energyassociated with a plasma system.

BACKGROUND

In a plasma-based system, plasma is generated within a plasma chamber toperform various operations, e.g., etching, cleaning, depositing, etc.,on a wafer. The plasma is monitored and controlled to controlperformance of the various operations. For example, the plasma ismonitored by using a bias compensation device to measure anelectrostatic chuck bias within the plasma chamber and by using avoltage probe at an output of an impedance matching circuit to measure aradio frequency (RF) voltage. The plasma is controlled by controlling anamount of RF power supplied to the plasma chamber.

However, the use of the bias compensation device and the voltage probeto monitor and control the performance of the operations may not providesatisfactory results. Moreover, the monitoring of wafer bias and the RFvoltage may be an expensive and time consuming operation.

It is in this context that embodiments described in the presentdisclosure arise.

SUMMARY

Embodiments of the disclosure provide apparatus, methods and computerprograms for using modeling to determine ion energy associated with aplasma system. It should be appreciated that the present embodiments canbe implemented in numerous ways, e.g., a process, an apparatus, asystem, a piece of hardware, or a method on a computer-readable medium.Several embodiments are described below.

In some embodiments, a method for determining ion energy is described.The method includes identifying a first complex voltage and currentmeasured at an output of a radio frequency (RF) generator when the RFgenerator is coupled to a plasma chamber via an impedance matchingcircuit. The impedance matching circuit has an input coupled to theoutput of the RF generator and an output coupled to an RF transmissionline. The method further includes generating an impedance matching modelbased on electrical components defined in the impedance matchingcircuit, the impedance matching model having an input and an output. Theinput of the impedance matching model receives the first complex voltageand current. The impedance matching model has one or more elements. Themethod includes propagating the first complex voltage and currentthrough the elements of the impedance matching model to determine asecond complex voltage and current. The method includes obtaining a peakvoltage; determining a wafer bias based on the second complex voltageand current, and determining an ion energy based on the wafer bias andthe peak voltage.

In various embodiments, a plasma system for determining an ion energy isdescribed. The plasma system includes an RF generator for generating anRF signal. The RF generator is associated with a voltage and currentprobe. The voltage and current probe is configured to measure a firstcomplex voltage and current at an output of the RF generator. The plasmasystem further includes an impedance matching circuit coupled to the RFgenerator and a plasma chamber coupled to the impedance matching circuitvia an RF transmission line. The impedance matching circuit has an inputcoupled to the output of the RF generator and an output coupled to theRF transmission line. The plasma system includes a processor coupled tothe RF generator. The processor is configured to identify the firstcomplex voltage and current and generate an impedance matching modelbased on electrical components defined in the impedance matchingcircuit. The impedance matching model has an input and an output, theinput of the impedance matching model receiving the first complexvoltage and current. The impedance matching model has one or moreelements. The processor is further configured to propagate the firstcomplex voltage and current through the elements of the impedancematching model to determine a second complex voltage and current, obtaina peak voltage, determine a wafer bias based on the second complexvoltage and current, and determine the ion energy based on the waferbias and the peak voltage.

A computer system for determining an ion energy is described. Thecomputer system includes a processor configured to identify a firstcomplex voltage and current measured at an output of an RF generatorwhen the RF generator is coupled to a plasma chamber via an impedancematching circuit. The impedance matching circuit has an input coupled tothe output of the RF generator and an output coupled to an RFtransmission line. The processor is further configured to generate animpedance matching model based on electrical components defined in theimpedance matching circuit. The impedance matching model has an inputand an output. The input of the impedance matching model receives thefirst complex voltage and current. The impedance matching model has oneor more elements. The processor is configured to propagate the firstcomplex voltage and current through the elements of the impedancematching model to determine a second complex voltage and current, obtaina peak voltage, determine a wafer bias based on the second complexvoltage and current, and determine the ion energy based on the waferbias and the peak voltage. The computer system includes a memory devicecoupled to the processor and the memory device is configured to storethe ion energy.

Some advantages of the above-described embodiments include determiningthe ion energy without a need to couple a voltage probe to an output ofan impedance matching circuit and without a need to use a biascompensation device to measure wafer bias. The voltage probe and thebias compensation circuit may be expensive to obtain. Comparatively, theion energy is determined without a need to couple the voltage probe tothe output of the impedance matching circuit and without a need to usethe bias compensation device. The nonuse of the voltage probe and thebias compensation device saves costs and time and effort associated withthe voltage probe and the bias compensation device.

Also, the voltage probe and the bias compensation circuit maymalfunction or may be unable to operate during manufacturing,processing, cleaning, etc. of a substrate. Comparatively, a voltage andcurrent probe that complies with a pre-set formula, is more reliable andis more accurate than the voltage probe is used in conjunction with amodel circuit to determine a radio frequency (RF) voltage and the RFvoltage is used to determine a wafer bias. The ion energy is determinedbased on the wafer bias and the RF voltage. The RF voltage and the waferbias determined using the voltage and current probe provides a betteraccuracy of ion energy than electrostatic chuck bias that is determinedbased on voltage measured by the voltage probe.

Other aspects will become apparent from the following detaileddescription, taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The embodiments may best be understood by reference to the followingdescription taken in conjunction with the accompanying drawings.

FIG. 1 is a block diagram of a system for determining a variable at anoutput of an impedance matching model, at an output of a portion of aradio frequency (RF) transmission model, and at an output of anelectrostatic chuck (ESC) model, in accordance with an embodimentdescribed in the present disclosure.

FIG. 2 is a flowchart of a method for determining a complex voltage andcurrent at the output of the RF transmission model portion, inaccordance with an embodiment described in the present disclosure.

FIG. 3A is a block diagram of a system used to illustrate an impedancematching circuit, in accordance with an embodiment described in thepresent disclosure.

FIG. 3B is a circuit diagram of an impedance matching model, inaccordance with an embodiment described in the present disclosure.

FIG. 4 is a diagram of a system used to illustrate an RF transmissionline, in accordance with an embodiment described in the presentdisclosure.

FIG. 5A is a block diagram of a system used to illustrate a circuitmodel of the RF transmission line, in accordance with an embodimentdescribed in the present disclosure.

FIG. 5B is a diagram of an electrical circuit used to illustrate atunnel and strap model of the RF transmission model, in accordance withan embodiment described in the present disclosure.

FIG. 5C is a diagram of an electrical circuit used to illustrate atunnel and strap model, in accordance with an embodiment described inthe present disclosure.

FIG. 6 is a diagram of an electrical circuit used to illustrate acylinder and ESC model, in accordance with an embodiment described inthe present disclosure.

FIG. 7 is a block diagram of a plasma system that includes filters usedto determine the variable, in accordance with an embodiment described inthe present disclosure.

FIG. 8A is a diagram of a system used to illustrate a model of thefilters to improve an accuracy of the variable, in accordance with anembodiment described in the present disclosure.

FIG. 8B is a diagram of a system used to illustrate a model of thefilters, in accordance with an embodiment described in the presentdisclosure.

FIG. 9 is a block diagram of a system for using a voltage and currentprobe to measure the variable at an output of an RF generator of thesystem of FIG. 1, in accordance with one embodiment described in thepresent disclosure.

FIG. 10 is a block diagram of a system in which the voltage and currentprobe and a communication device are located outside the RF generator,in accordance with an embodiment described in the present disclosure.

FIG. 11 is a block diagram of a system in which values of the variabledetermined using the system of FIG. 1 are used, in accordance with anembodiment described in the present disclosure.

FIG. 12A is a diagram of a graph that illustrates a correlation betweena variable that is measured at a node within the system of FIG. 1 byusing a probe and a variable that is determined using the method of FIG.2 when an x MHz RF generator is on, in accordance with an embodimentdescribed in the present disclosure.

FIG. 12B is a diagram of a graph that illustrates a correlation betweena variable that is measured at a node within the system of FIG. 1 byusing a probe and a variable that is determined using the method of FIG.2 when a y MHz RF generator is on, in accordance with an embodimentdescribed in the present disclosure.

FIG. 12C is a diagram of a graph that illustrates a correlation betweena variable that is measured at a node within the system of FIG. 1 byusing a probe and a variable that is determined using the method of FIG.2 when a z MHz RF generator is on, in accordance with one embodimentdescribed in the present disclosure.

FIG. 13 is a flowchart of a method for determining wafer bias at a modelnode of the impedance matching model, the RF transmission model, or theESC model, in accordance with an embodiment described in the presentdisclosure.

FIG. 14 is a state diagram illustrating a wafer bias generator used togenerate a wafer bias, in accordance with an embodiment described in thepresent disclosure.

FIG. 15 is a flowchart of a method for determining a wafer bias at apoint along a path between the impedance matching model and the ESCmodel, in accordance with an embodiment described in the presentdisclosure.

FIG. 16 is a block diagram of a system for determining a wafer bias at anode of a model, in accordance with an embodiment described in thepresent disclosure.

FIG. 17 is a flowchart of a method for determining a wafer bias at amodel node of the system of FIG. 1, in accordance with an embodimentdescribed in the present disclosure.

FIG. 18 is a block diagram of a system that is used to illustrateadvantages of determining wafer bias by using the method of FIG. 13,FIG. 15, or FIG. 17 instead of by using a voltage probe, in accordancewith an embodiment described in the present disclosure.

FIG. 19A show embodiments of graphs to illustrate a correlation betweena variable that is measured at a node of the plasma system of FIG. 1 byusing a voltage probe and a variable at a corresponding model nodeoutput determined using the method of FIG. 2, 13, 15, or 17 when the yand z MHz RF generators are on, in accordance with an embodimentdescribed in the present disclosure.

FIG. 19B show embodiments of graphs to illustrate a correlation betweena variable that is measured at a node of the plasma system of FIG. 1 byusing a voltage probe and a variable at a corresponding model nodeoutput determined using the method of FIG. 2, 13, 15, or 17 when the xand z MHz RF generators are on, in accordance with an embodimentdescribed in the present disclosure.

FIG. 19C show embodiments of graphs to illustrate a correlation betweena variable that is measured at a node of the plasma system of FIG. 1 byusing a voltage probe and a variable at a corresponding model nodeoutput determined using the method of FIG. 2, 13, 15, or 17 when the xand y MHz RF generators are on, in accordance with an embodimentdescribed in the present disclosure.

FIG. 20A is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model wafer bias thatis determined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the x MHz RF generator is on, in accordance with anembodiment described in the present disclosure.

FIG. 20B is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model bias that isdetermined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the y MHz RF generator is on, in accordance with oneembodiment described in the present disclosure.

FIG. 20C is a diagram of embodiments of graphs used to illustrate acorrelation between a wired wafer bias measured using a sensor tool, amodel bias that is determined using the method of FIG. 13, 15, or 17 andan error in the model bias when the z MHz RF generator is on, inaccordance with one embodiment described in the present disclosure.

FIG. 20D is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model bias that isdetermined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the x and y MHz RF generators are on, in accordance withan embodiment described in the present disclosure.

FIG. 20E is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model bias that isdetermined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the x and z MHz RF generators are on, in accordance withan embodiment described in the present disclosure.

FIG. 20F is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model bias that isdetermined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the y and z MHz RF generators are on, in accordance withan embodiment described in the present disclosure.

FIG. 20G is a diagram of graphs used to illustrate a correlation betweena wired wafer bias measured using a sensor tool, a model bias that isdetermined using the method of FIG. 13, 15, or 17 and an error in themodel bias when the x, y, and z MHz RF generators are on, in accordancewith an embodiment described in the present disclosure.

FIG. 21 is a block diagram of a host system of the system of FIG. 1, inaccordance with an embodiment described in the present disclosure.

FIG. 22 is a diagram of a function used to illustrate determination ofion energy from a wafer bias and peak magnitude.

DETAILED DESCRIPTION

The following embodiments describe systems and methods for using a modelto determine an ion energy associated with a plasma system. It will beapparent that the present embodiments may be practiced without some orall of these specific details. In other instances, well known processoperations have not been described in detail in order not tounnecessarily obscure the present embodiments.

FIG. 1 is a block diagram of an embodiment of a system 126 fordetermining a variable at an output of an impedance matching model 104,at an output, e.g., a model node N1 m, of a portion 173 of an RFtransmission model 161, which is a model of an RF transmission line 113,and at an output, e.g., a model node N6 m, of an electrostatic chuck(ESC) model 125. Examples of a variable include complex voltage, complexcurrent, complex voltage and current, complex power, ion energy, waferbias, etc. The RF transmission line 113 has an output, e.g., a node N2.A voltage and current (VI) probe 110 measures a complex voltage andcurrent V_(xMHz), I_(xMHZ), and ϕ_(xMHz), e.g., a first complex voltageand current, at an output, e.g., a node N3, of an x MHz RF generator. Itshould be noted that V_(xMHz) represents a voltage magnitude, I_(xMHz)represents a current magnitude, and ϕ_(xMHz) represents a phase betweenV_(xMHz) and I_(xMHz). The impedance matching model 104 has an output,e.g., a model node N4 m.

Moreover, a voltage and current probe 111 measures a complex voltage andcurrent V_(yMHz), I_(yMHz), and ϕ_(yMHz) at an output, e.g., a node N5,of a y MHz RF generator. It should be noted that V_(yMHz) represents avoltage magnitude, I_(yMHz) represents a current magnitude, and ϕ_(yMHz)represents a phase between V_(yMHz) and I_(yMHz).

In some embodiments, a node is an input of a device, an output of adevice, or a point within the device. A device, as used herein, isdescribed below.

In various embodiments, a voltage magnitude includes a zero-to-peakmagnitude, or a peak-to-peak magnitude, or a root mean square (RMS)magnitude, which is of one or more radio frequency values of an RFsignal. In some embodiments, a current magnitude includes a zero-to-peakmagnitude, or a peak-to-peak magnitude, or an RMS magnitude, which is ofone or more radio frequency values of an RF signal. In severalembodiments, a power magnitude is a product of a voltage magnitude, acurrent magnitude, and a phase between the current magnitude and thevoltage magnitude.

Examples of x MHz include 2 MHz, 27 MHz, and 60 MHz. Examples of y MHzinclude 2 MHz, 27 MHz, and 60 MHz. The x MHz is different than y MHz.For example, when x MHz is 2 MHz, y MHz is 27 MHz or 60 MHz. When x MHzis 27 MHz, y MHz is 60 MHz.

An example of each voltage and current probe 110 and 111 includes avoltage and current probe that complies with a pre-set formula. Anexample of the pre-set formula includes a standard that is followed byan Association, which develops standards for sensors. Another example ofthe pre-set formula includes a National Institute of Standards andTechnology (NIST) standard. As an illustration, the voltage and currentprobe 110 or 111 is calibrated according to NIST standard. In thisillustration, the voltage and current probe 110 or 111 is coupled withan open circuit, a short circuit, or a known load to calibrate thevoltage and current probe 110 or 111 to comply with the NIST standard.The voltage and current probe 110 or 111 may first be coupled with theopen circuit, then with the short circuit, and then with the known loadto calibrate the voltage and current probe 110 based on NIST standard.The voltage and current probe 110 or 111 may be coupled to the knownload, the open circuit, and the short circuit in any order to calibratethe voltage and current probe 110 or 111 according to NIST standard.Examples of a known load include a 50 ohm load, a 100 ohm load, a 200ohm load, a static load, a direct current (DC) load, a resistor, etc. Asan illustration, each voltage and current probe 110 and 111 iscalibrated according NIST-traceable standards.

The voltage and current probe 110 is coupled to the output, e.g., thenode N3, of the x MHz RF generator. The output, e.g., the node N3, ofthe x MHz RF generator is coupled to an input 153 of an impedancematching circuit 114 via a cable 150. Moreover, the voltage and currentprobe 111 is coupled to the output, e.g., the node N5, of the y MHz RFgenerator. The output, e.g., the node N5, of the y MHz RF generator iscoupled to another input 155 of the impedance matching circuit 114 via acable 152.

An output, e.g., a node N4, of the impedance matching circuit 114 iscoupled to an input of the RF transmission line 113. The RF transmissionline 113 includes a portion 169 and another portion 195. An input of theportion 169 is an input of the RF transmission line 113. An output,e.g., a node N1, of the portion 169 is coupled to an input of theportion 195. An output, e.g., the node N2, of the portion 195 is coupledto the plasma chamber 175. The output of the portion 195 is the outputof the RF transmission line 113. An example of the portion 169 includesan RF cylinder and an RF strap. The RF cylinder is coupled to the RFstrap. An example of the portion 195 includes an RF rod and/or asupport, e.g., a cylinder, etc., for supporting the plasma chamber 175.

The plasma chamber 175 includes an electrostatic chuck (ESC) 177, anupper electrode 179, and other parts (not shown), e.g., an upperdielectric ring surrounding the upper electrode 179, an upper electrodeextension surrounding the upper dielectric ring, a lower dielectric ringsurrounding a lower electrode of the ESC 177, a lower electrodeextension surrounding the lower dielectric ring, an upper plasmaexclusion zone (PEZ) ring, a lower PEZ ring, etc. The upper electrode179 is located opposite to and facing the ESC 177. A work piece 131,e.g., a semiconductor wafer, etc., is supported on an upper surface 183of the ESC 177. The upper surface 183 includes an output N6 of the ESC177. The work piece 131 is placed on the output N6. Various processes,e.g., chemical vapor deposition, cleaning, deposition, sputtering,etching, ion implantation, resist stripping, etc., are performed on thework piece 131 during production. Integrated circuits, e.g., applicationspecific integrated circuit (ASIC), programmable logic device (PLD),etc. are developed on the work piece 131 and the integrated circuits areused in a variety of electronic items, e.g., cell phones, tablets, smartphones, computers, laptops, networking equipment, etc. Each of the lowerelectrode and the upper electrode 179 is made of a metal, e.g.,aluminum, alloy of aluminum, copper, etc.

In one embodiment, the upper electrode 179 includes a hole that iscoupled to a central gas feed (not shown). The central gas feed receivesone or more process gases from a gas supply (not shown). Examples of aprocess gases include an oxygen-containing gas, such as O₂. Otherexamples of a process gas include a fluorine-containing gas, e.g.,tetrafluoromethane (CF₄), sulfur hexafluoride (SF₆), hexafluoroethane(C₂F₆), etc. The upper electrode 179 is grounded. The ESC 177 is coupledto the x MHz RF generator and the y MHz RF generator via the impedancematching circuit 114.

When the process gas is supplied between the upper electrode 179 and theESC 177 and when the x MHz RF generator and/or the y MHz RF generatorsupplies RF signals via the impedance matching circuit 114 and the RFtransmission line 113 to the ESC 177, the process gas is ignited togenerate plasma within the plasma chamber 175.

When the x MHz RF generator generates and provides an RF signal via thenode N3, the impedance matching circuit 114, and the RF transmissionline 113 to the ESC 177 and when the y MHz generator generates andprovides an RF signal via the node N5, the impedance matching circuit114, and the RF transmission line 113 to the ESC 177, the voltage andcurrent probe 110 measures the complex voltage and current at the nodeN3 and the voltage and current probe 111 measures the complex voltageand current at the node N5.

The complex voltages and currents measured by the voltage and currentprobes 110 and 111 are provided via corresponding communication devices185 and 189 from the corresponding voltage and current probes 110 and111 to a storage hardware unit (HU) 162 of a host system 130 forstorage. For example, the complex voltage and current measured by thevoltage and current probe 110 is provided via the communication device185 and a cable 191 to the host system 130 and the complex voltage andcurrent measured by the voltage and current probe 111 is provided viathe communication device 189 and a cable 193 to the host system 130.Examples of a communication device include an Ethernet device thatconverts data into Ethernet packets and converts Ethernet packets intodata, an Ethernet for Control Automation Technology (EtherCAT) device, aserial interface device that transfers data in series, a parallelinterface device that transfers data in parallel, a Universal Serial Bus(USB) interface device, etc.

Examples of the host system 130 include a computer, e.g., a desktop, alaptop, a tablet, etc. As an illustration, the host system 130 includesa processor and the storage HU 162. As used herein, a processor may be acentral processing unit (CPU), a microprocessor, an application specificintegrated circuit (ASIC), a programmable logic device (PLD), etc.Examples of the storage HU include a read-only memory (ROM), a randomaccess memory (RAM), or a combination thereof. The storage HU may be aflash memory, a redundant array of storage disks (RAID), a hard disk,etc.

The impedance matching model 104 is stored within the storage HU 162.The impedance matching model 104 has similar characteristics, e.g.,capacitances, inductances, complex power, complex voltage and currents,etc., as that of the impedance matching circuit 114. For example, theimpedance matching model 104 has the same number of capacitors and/orinductors as that within the impedance matching circuit 114, and thecapacitors and/or inductors are connected with each other in the samemanner, e.g., serial, parallel, etc. as that within the impedancematching circuit 114. To provide an illustration, when the impedancematching circuit 114 includes a capacitor coupled in series with aninductor, the impedance matching model 104 also includes the capacitorcoupled in series with the inductor.

As an example, the impedance matching circuit 114 includes one or moreelectrical components and the impedance matching model 104 includes adesign, e.g., a computer-generated model, of the impedance matchingcircuit 114. The computer-generated model may be generated by aprocessor based upon input signals received from a user via an inputhardware unit. The input signals include signals regarding whichelectrical components, e.g., capacitors, inductors, etc., to include ina model and a manner, e.g., series, parallel, etc., of coupling theelectrical components with each other. As another example, the impedancecircuit 114 includes hardware electrical components and hardwareconnections between the electrical components and the impedance matchingmodel 104 include software representations of the hardware electricalcomponents and of the hardware connections. As yet another example, theimpedance matching model 104 is designed using a software program andthe impedance matching circuit 114 is made on a printed circuit board.As used herein, electrical components may include resistors, capacitors,inductors, connections between the resistors, connections between theinductors, connections between the capacitors, and/or connectionsbetween a combination of the resistors, inductors, and capacitors.

Similarly, a cable model 163 and the cable 150 have similarcharacteristics, and a cable model 165 and the cable 152 has similarcharacteristics. As an example, an inductance of the cable model 163 isthe same as an inductance of the cable 150. As another example, thecable model 163 is a computer-generated model of the cable 150 and thecable model 165 is a computer-generated model of the cable 152.

Similarly, an RF transmission model 161 and the RF transmission line 113have similar characteristics. For example, the RF transmission model 161has the same number of resistors, capacitors and/or inductors as thatwithin the RF transmission line 113, and the resistors, capacitorsand/or inductors are connected with each other in the same manner, e.g.,serial, parallel, etc. as that within the RF transmission line 113. Tofurther illustrate, when the RF transmission line 113 includes acapacitor coupled in parallel with an inductor, the RF transmissionmodel 161 also includes the capacitor coupled in parallel with theinductor. As yet another example, the RF transmission line 113 includesone or more electrical components and the RF transmission model 161includes a design, e.g., a computer-generated model, of the RFtransmission line 113.

In some embodiments, the RF transmission model 161 is acomputer-generated impedance transformation involving computation ofcharacteristics, e.g., capacitances, resistances, inductances, acombination thereof, etc., of elements, e.g., capacitors, inductors,resistors, a combination thereof, etc., and determination ofconnections, e.g., series, parallel, etc., between the elements.

Based on the complex voltage and current received from the voltage andcurrent probe 110 via the cable 191 and characteristics, e.g.,capacitances, inductances, etc., of elements, e.g., inductors,capacitors, etc., within the impedance matching model 104, the processorof the host system 130 calculates a complex voltage and current V, I,and ϕ, e.g., a second complex voltage and current, at the output, e.g.,the model node N4 m, of the impedance matching model 104. The complexvoltage and current at the model node N4 m is stored in the storage HU162 and/or another storage HU, e.g., a compact disc, a flash memory,etc., of the host system 130. The complex V, I, and ϕ includes a voltagemagnitude V, a current magnitude I, and a phase ϕ between the voltageand current.

The output of the impedance matching model 104 is coupled to an input ofthe RF transmission model 161, which is stored in the storage hardwareunit 162. The impedance matching model 104 also has an input, e.g., anode N3 m, which is used to receive the complex voltage and currentmeasured at the node N3.

The RF transmission model 161 includes the portion 173, another portion197, and an output N2 m, which is coupled via the ESC model 125 to themodel node N6 m. The ESC model 125 is a model of the ESC 177. Forexample, the ESC model 125 has similar characteristics as that of theESC 177. For example, the ESC model 125 has the same inductance,capacitance, resistance, or a combination thereof as that of the ESC177.

An input of the portion 173 is the input of the RF transmission model161. An output of the portion 173 is coupled to an input of the portion197. The portion 172 has similar characteristics as that of the portion169 and the portion 197 has similar characteristics as that of theportion 195.

Based on the complex voltage and current measured at the model node N4m, the processor of the host system 130 calculates a complex voltage andcurrent V, I, and ϕ, e.g., a third complex voltage and current, at theoutput, e.g., the model node N1 m, of the portion 173 of the RFtransmission model 161. The complex voltage and current determined atthe model node N1 m is stored in the storage HU 162 and/or anotherstorage HU, e.g., a compact disc, a flash memory, etc., of the hostsystem 130.

In several embodiments, instead of or in addition to determining thethird complex voltage and current, the processor of the host system 130computes a complex voltage and current, e.g., an intermediate complexvoltage and current V, I, and ϕ, at a point, e.g., a node, etc., withinthe portion 173 based on the complex voltage and current at the outputof the impedance matching model 104 and characteristics of elementsbetween the input of the RF transmission model 161 and the point withinthe portion 173.

In various embodiments, instead of or in addition to determining thethird complex voltage and current, the processor of the host system 130computes a complex voltage and current, e.g., an intermediate complexvoltage and current V, I, and ϕ, at a point, e.g., a node, etc., withinthe portion 197 based on the complex voltage and current at the outputof the impedance matching model 104 and characteristics of elementsbetween the input of the RF transmission model 161 and the point withinthe portion 197.

It should be noted that in some embodiments, the complex voltage andcurrent at the output of the impedance matching model 104 is calculatedbased on the complex voltage and current at the output of the x MHz RFgenerator, characteristics of elements the cable model 163, andcharacteristics of the impedance matching model 104.

It should further be noted that although two generators are showncoupled to the impedance matching circuit 114, in one embodiment, anynumber of RF generators, e.g., a single generator, three generators,etc., are coupled to the plasma chamber 175 via an impedance matchingcircuit. For example, a 2 MHz generator, a 27 MHz generator, and a 60MHz generator may be coupled to the plasma chamber 175 via an impedancematching circuit. For example, although the above-described embodimentsare described with respect to using complex voltage and current measuredat the node N3, in various embodiments, the above-described embodimentsmay also use the complex voltage and current measured at the node N5.

FIG. 2 is a flowchart of an embodiment of a method 102 for determiningthe complex voltage and current at the output of the RF transmissionmodel portion 173 (FIG. 1). The method 102 is executed by the processorof the host system 130 (FIG. 1). In an operation 106, the complexvoltage and current, e.g., the first complex voltage and current,measured at the node N3 is identified from within the storage HU 162(FIG. 1). For example, it is determined that the first complex voltageand current is received from the voltage and current probe 110 (FIG. 1).As another example, based on an identity, of the voltage and currentprobe 110, stored within the storage HU 162 (FIG. 1), it is determinedthat the first complex voltage and current is associated with theidentity.

Furthermore, in an operation 107, the impedance matching model 104(FIG. 1) is generated based on electrical components of the impedancematching circuit 114 (FIG. 1). For example, connections betweenelectrical components of the impedance matching circuit 114 andcharacteristics of the electrical components are provided to theprocessor of the host system 130 by the user via an input hardware unitthat is coupled with the host system 130. Upon receiving the connectionsand the characteristics, the processor generates elements that have thesame characteristics as that of electrical components of the impedancematching circuit 114 and generates connections between the elements thathave the same connections as that between the electrical components.

The input, e.g., the node N3 m, of the impedance matching model 163receives the first complex voltage and current. For example, theprocessor of the host system 130 accesses, e.g., reads, etc., from thestorage HU 162 the first complex voltage and current and provides thefirst complex voltage and current to the input of the impedance matchingmodel 104 to process the first complex voltage and current.

In an operation 116, the first complex voltage and current is propagatedthrough one or more elements of the impedance matching model 104(FIG. 1) from the input, e.g., the node N3 m (FIG. 1), of the impedancematching model 104 to the output, e.g., the node N4 m (FIG. 1), of theimpedance matching model 104 to determine the second complex voltage andcurrent, which is at the output of the impedance matching model 104. Forexample, with reference to FIG. 3B, when the 2 MHz RF generator is on,e.g., operational, powered on, coupled to the devices, such as, forexample, the impedance matching circuit 104, of the plasma system 126,etc., a complex voltage and current Vx1, Ix1, and ϕx1, e.g., anintermediate complex voltage and current, which includes the voltagemagnitude Vx1, the current magnitude Ix1, and the phase ϕx1 between thecomplex voltage and current, at a node 251, e.g., an intermediate node,is determined based on a capacitance of a capacitor 253, based on acapacitance of a capacitor C5, and based on the first complex voltageand current that is received at an input 255. Moreover, a complexvoltage and current Vx2, Ix2, and ϕx2 at a node 257 is determined basedon the complex voltage and current Vx1, Ix1, and ϕx1, and based on aninductance of an inductor L3. The complex voltage and current Vx2, Ix2,and ϕx2 includes the voltage magnitude Vx2, the current magnitude Ix2,and the phase ϕx2 between the voltage and current. When the 27 MHz RFgenerator and the 60 MHz RF generator are off, e.g., nonoperational,powered off, decoupled from the impedance matching circuit 104, etc., acomplex voltage and current V2, I2, and ϕ2 is determined to be thesecond complex voltage and current at an output 259, which is an exampleof the output, e.g., the model node N4 m (FIG. 1), of the impedancematching model 104 (FIG. 1). The complex voltage and current V2, I2, andϕ2 is determined based on the complex voltage and current Vx2, Ix2, andϕx2 and an inductor of an inductor L2. The complex voltage and currentV2, I2, and ϕ2 includes the voltage magnitude V2, the current magnitudeI2, and the phase ϕ2 between the voltage and current.

Similarly, when 27 MHz RF generator is on and the 2 MHz and the 60 MHzRF generators are off, a complex voltage and current V27, I27, and ϕ27at the output 259 is determined based on a complex voltage and currentreceived at a node 261 and characteristics of an inductor LPF2, acapacitor C3, a capacitor C4, and an inductor L2. The complex voltageand current V27, I27, and ϕ27 includes the voltage magnitude V27, thecurrent magnitude I27, and the phase ϕ27 between the voltage andcurrent. The complex voltage and current received at the node 261 is thesame as the complex voltage and current measured at the node N5 (FIG.1). When both the 2 MHz and 27 MHz RF generators are on and the 60 MHzRF generator is off, the complex voltages and currents V2, I2, ϕ2, V27,I27, and ϕ27 are an example of the second complex voltage and current.Moreover, similarly, when the 60 MHz RF generator is on and the 2 and 27MHz RF generators are off, a complex voltage and current V60, I60, andϕ60 at the output 259 is determined based on a complex voltage andcurrent received at a node 265 and characteristics of an inductor LPF1,a capacitor C1, a capacitor C2, an inductor L4, a capacitor 269, and aninductor L1. The complex voltage and current V60, I60, and ϕ60 includesthe voltage magnitude V60, the current magnitude I60, and the phase ϕ60between the voltage and current. When the 2 MHz, 27 MHz, and the 60 MHzRF generators are on, the complex voltages and currents V2, I2, ϕ2, V27,I27, ϕ27, V60, I60, and ϕ60 are an example of the second complex voltageand current.

In an operation 117, the RF transmission model 161 (FIG. 1) is generatedbased on the electrical components of the RF transmission line 113 (FIG.1). For example, connections between electrical components of the RFtransmission line 113 and characteristics of the electrical componentsare provided to the processor of the host system 130 by the user via aninput device that is coupled with the host system 130. Upon receivingthe connections and the characteristics, the processor generateselements that have the same characteristics as that of electricalcomponents of the RF transmission line 113 and generates connectionsbetween the elements that are the same as that between the electricalcomponents.

In an operation 119, the second complex voltage and current ispropagated through one or more elements of the RF transmission modelportion 173 from the input of the RF transmission model 113 to theoutput, e.g., the model node N1 m (FIG. 1), of the RF transmission modelportion 173 to determine the third complex voltage and current at theoutput of the RF transmission model portion 173. For example, withreference to FIG. 5B, when the 2 MHz RF generator is on and the 27 and60 MHz RF generators are off, a complex voltage and current Vx4, Ix4,and ϕ4, e.g., an intermediate complex voltage and current, at a node293, e.g., an intermediate node, is determined based on an inductance ofan inductor Ltunnel, based on a capacitance of a capacitor Ctunnel, andbased on the complex voltage and current V2, I2, and ϕ2 (FIG. 3B), whichis an example of the second complex voltage and current. It should benoted that Ltunnel is an inductance of a computer-generated model of anRF tunnel and Ctunnel is a capacitance of the RF tunnel model. Moreover,a complex voltage and current V21, I21, and ϕ21 at an output 297 of atunnel and strap model 210 is determined based on the complex voltageand current Vx4, Ix4, and ϕx4, and based on an inductance of an inductorLstrap. The output 297 is an example of the output, e.g., the model nodeN1 m (FIG. 1), of the portion 173 (FIG. 1). It should be noted thatLstrap is an inductance of a computer-generated model of the RF strap.When the 2 MHz RF generator is on and the 27 and 60 MHz RF generatorsare off, the complex voltage and current V21, I21, and ϕ21 is determinedto be the third complex voltage and current at the output 297.

Similarly, when the 27 MHz RF generator is on and the 2 and 60 MHz RFgenerators are off, a complex voltage and current V271, I271, and ϕ271at the output 297 is determined based on the complex voltage and currentV27, I27, ϕ27 (FIG. 3B) at the output 259 and characteristics of theinductor Ltunnel, the capacitor Ctunnel, and the inductor Lstrap. Whenboth the 2 MHz and 27 MHz RF generators are on and the 60 MHz RFgenerator is off, the complex voltages and currents V21, I21, ϕ21, V271,I271, and ϕ271 are an example of the third complex voltage and current.

Moreover, similarly, when the 60 MHz RF generator is powered on and the2 and 27 MHz RF generators are powered off, a complex voltage andcurrent V601, I601, and ϕ601 at the output 297 is determined based onthe complex voltage and current V60, I60, and ϕ60 (FIG. 3B) received ata node 259 and characteristics of the inductor Ltunnel, the capacitorCtunnel, and the inductor Lstrap. When the 2 MHz, 27 MHz, and the 60 MHzRF generators are on, the complex voltages and currents V21, I21, ϕ21,V271, I271, ϕ271, V601, I601, and ϕ601 are an example of the thirdcomplex voltage and current. The method 102 ends after the operation119.

FIG. 3A is a block diagram of an embodiment of a system 123 used toillustrate an impedance matching circuit 122. The impedance matchingcircuit 122 is an example of the impedance matching circuit 114 (FIG.1). The impedance matching circuit 122 includes series connectionsbetween electrical components and/or parallel connections betweenelectrical components.

FIG. 3B is a circuit diagram of an embodiment of an impedance matchingmodel 172. The impedance matching model 172 is an example of theimpedance matching model 104 (FIG. 1). As shown, the impedance matchingmodel 172 includes capacitors having capacitances C1 thru C9, inductorshaving inductances LPF1, LPF2, and L1 thru L4. It should be noted thatthe manner in which the inductors and/or capacitors are coupled witheach other in FIG. 3B is an example. For example, the inductors and/orcapacitors shown in FIG. 3B can be coupled in a series and/or parallelmanner with each other. Also, in some embodiments, the impedancematching model 172 includes a different number of capacitors and/or adifferent number of inductors than that shown in FIG. 3B.

FIG. 4 is a diagram of an embodiment of a system 178 used to illustratean RF transmission line 181, which is an example of the RF transmissionline 113 (FIG. 1). The RF transmission line 181 includes a cylinder 148,e.g., a tunnel. Within a hollow of the cylinder 148 lies an insulator151 and an RF rod 142. A combination of the cylinder 148 and the RF rod142 is an example of the portion 169 (FIG. 1) of the RF transmissionline 113 (FIG. 1). The RF transmission line 181 is bolted via bolts B1,B2, B3, and B4 with the impedance matching circuit 114. In oneembodiment, the RF transmission line 181 is bolted via any number ofbolts with the impedance matching circuit 114. In some embodiments,instead of or in addition to bolts, any other form of attachment, e.g.,glue, screws, etc., is used to attach the RF transmission line 181 tothe impedance matching circuit 114.

The RF transmission rod 142 is coupled with the output of the impedancematching circuit 114. Also, an RF strap 144, also known as RF spoon, iscoupled with the RF rod 142 and with an RF rod 199, a portion of whichis located within a support 146, e.g., a cylinder. The support 146 thatincludes the RF rod 199 is an example of the portion 195 (FIG. 1). In anembodiment, a combination of the cylinder 148, the RF rod 142, the RFstrap 144, the support 146 and the RF rod 199 forms the RF transmissionline 181, which is an example of the RF transmission line 113 (FIG. 1).The support 146 provides support to the plasma chamber. The support 146is attached to the ESC 177 of the plasma chamber. An RF signal issupplied from the x MHz generator via the cable 150, the impedancematching circuit 114, the RF rod 142, the RF strap 144, and the RF rod199 to the ESC 177.

In one embodiment, the ESC 177 includes a heating element and anelectrode on top of the heating element. In an embodiment, the ESC 177includes a heating element and the lower electrode. In one embodiment,the ESC 177 includes the lower electrode and a heating element, e.g.,coil wire, etc., embedded within holes formed within the lowerelectrode. In some embodiments, the electrode is made of a metal, e.g.,aluminum, copper, etc. It should be noted that the RF transmission line181 supplies an RF signal to the lower electrode of the ESC 177.

FIG. 5A is a block diagram of an embodiment of a system 171 used toillustrate a circuit model 176 of the RF transmission line 113 (FIG. 1).For example, the circuit model 176 includes inductors and/or capacitors,connections between the inductors, connections between the capacitors,and/or connections between the inductors and the capacitors. Examples ofconnections include series and/or parallel connections. The circuitmodel 176 is an example of the RF transmission model 161 (FIG. 1).

FIG. 5B is a diagram of an embodiment of an electrical circuit 180 usedto illustrate the tunnel and strap model 210, which is an example of theportion 173 (FIG. 1) of the RF transmission model 161 (FIG. 1). Theelectrical circuit 180 includes the impedance matching model 172 and thetunnel and strap model 210. The tunnel and strap model 210 includesinductors Ltunnel and Lstrap and a capacitor Ctunnel. It should be notedthat the inductor Ltunnel represents an inductance of the cylinder 148(FIG. 4) and the RF rod 142 and the capacitor Ctunnel represents acapacitance of the cylinder 148 and the RF rod 142. Moreover, theinductor Lstrap represents an inductance of the RF strap 144 (FIG. 4).

In an embodiment, the tunnel and strap model 210 includes any number ofinductors and/or any number of capacitors. In this embodiment, thetunnel and strap model 210 includes any manner, e.g., serial, parallel,etc. of coupling a capacitor to another capacitor, coupling a capacitorto an inductor, and/or coupling an inductor to another inductor.

FIG. 5C is a diagram of an embodiment of an electrical circuit 300 usedto illustrate a tunnel and strap model 302, which is an example of theportion 173 (FIG. 1) of the RF transmission model 161 (FIG. 1). Thetunnel and strap model 302 is coupled via the output 259 to theimpedance matching model 172. The tunnel and strap model 302 includesinductors having inductances 20 nanoHenry (nH) and capacitors havingcapacitances of 15 picoFarads (pF), 31 pF, 15.5 pF, and 18.5 pF. Thetunnel and strap model 302 is coupled via a node 304 to an RF cylinder,which is coupled to the ESC 177 (FIG. 1). The RF cylinder is an exampleof the portion 195 (FIG. 1).

It should be noted that in some embodiments, the inductors andcapacitors of the tunnel and strap model 302 have other values. Forexample, the 20 nH inductors have an inductance ranging between 15 and20 nH or between 20 and 25 nH. As another example, two or more of theinductors of the tunnel and strap model 302 have difference inductances.As yet another example, the 15 pF capacitor has a capacitance rangingbetween 8 pF and 25 pF, the 31 pF capacitor has a capacitance rangingbetween 15 pF and 45 pF, the 15.5 pF capacitor has a capacitance rangingbetween 9 pF and 20 pF, and the 18.5 pF capacitor has a capacitanceranging between 10 pF and 27 pF.

In various embodiments, any number of inductors are included in thetunnel and strap model 302 and any number of capacitors are included inthe tunnel and strap model 302.

FIG. 6 is a diagram of an embodiment of an electrical circuit 310 usedto illustrate a cylinder and ESC model 312, which is a combination of aninductor 313 and a capacitor 316. The cylinder and ESC model 312includes a cylinder model and an ESC model, which is an example of theESC model 125 (FIG. 1). The cylinder model is an example of the portion197 (FIG. 1) of the RF transmission model 161 (FIG. 1). The cylinder andESC model 312 has similar characteristics as that of a combination ofthe portion 195 and the ESC 177 (FIG. 1). For example, the cylinder andESC model 312 has the same resistance as that of a combination of theportion 195 and the ESC 177. As another example, the cylinder and ESCmodel 312 has the same inductance as that of a combination of theportion 195 and the ESC 177. As yet another example, the cylinder andESC model 312 has the same capacitance as that of a combination of theportion 195 and the ESC 177. As yet another example, the cylinder andESC model 312 has the same inductance, resistance, capacitance, or acombination thereof, as that of a combination of the portion 195 and theESC 177.

The cylinder and ESC model 312 is coupled via a node 318 to the tunneland strap model 302. The node 318 is an example of the model node N1 m(FIG. 1).

It should be noted that in some embodiments, an inductor having aninductance other than the 44 milliHenry (mH) is used in the cylinder andESC model 312. For example, an inductor having an inductance rangingfrom 35 mH to 43.9 mH or from 45.1 mH too 55 mH is used. In variousembodiments, a capacitor having a capacitance other than 550 pF is used.For example, instead of the 550 pF capacitor, a capacitor having acapacitance ranging between 250 and 550 pF or between 550 and 600 pF isused.

The processor of the host system 130 (FIG. 1) calculates a combinedimpedance, e.g., total impedance, etc., of a combination of the model172, the tunnel and strap model 302, and the cylinder and ESC model 312.The combined impedance and complex voltage and current determined at themodel node 318 are used as inputs by the processor of the host system130 to calculate a complex voltage and impedance at the node N6 m. Itshould be noted that an output of the cylinder and ESC model 312 is themodel node N6 m.

FIG. 7 is a block diagram of an embodiment of a system 200 that is usedto determine a variable. The system 200 includes a plasma chamber 135,which further includes an ESC 201 and has an input 285. The plasmachamber 135 is an example of the plasma chamber 175 (FIG. 1) and the ESC201 is an example of the ESC 177 (FIG. 1). The ESC 201 includes aheating element 198. Also, the ESC 201 is surrounded by an edge ring(ER) 194. The ER 194 includes a heating element 196. In an embodiment,the ER 194 facilitates a uniform etch rate and reduced etch rate driftnear an edge of the work piece 131 that is supported by the ESC 201.

A power supply 206 provides power to the heating element 196 via afilter 208 to heat the heating element 196 and a power supply 204provides power to the heating element 198 via a filter 202 to heat theheating element 198. In an embodiment, a single power supply providespower to both the heating elements 196 and 198. The filter 208 filtersout predetermined frequencies of a power signal that is received fromthe power supply 206 and the filter 202 filters out predeterminedfrequencies of a power signal that is received from the power supply204.

The heating element 198 is heated by the power signal received from thepower supply 204 to maintain an electrode of the ESC 201 at a desirabletemperature to further maintain an environment within the plasma chamber135 at a desirable temperature. Moreover, the heating element 196 isheated by the power signal received from the power supply 206 tomaintain the ER 194 at a desirable temperature to further maintain anenvironment within the plasma chamber 135 at a desirable temperature.

It should be noted that in an embodiment, the ER 194 and the ESC 201include any number of heating elements and any type of heating elements.For example, the ESC 201 includes an inductive heating element or ametal plate. In one embodiment, each of the ESC 201 and the ER 194includes one or more cooling elements, e.g., one or more tubes thatallow passage of cold water, etc., to maintain the plasma chamber 135 ata desirable temperature.

It should further be noted that in one embodiment, the system 200includes any number of filters. For example, the power supplies 204 and206 are coupled to the ESC 201 and the ER 194 via a single filter.

FIG. 8A is a diagram of an embodiment of a system 217 used to illustratea model of the filters 202 and 208 (FIG. 7) to improve an accuracy ofthe variable. The system 217 includes the tunnel and strap model 210that is coupled via a cylinder model 211 to a model 216, which includescapacitors and/or inductors and connections therebetween of the filters202 and 208. The model 216 is stored within the storage HU 162 (FIG. 1)and/or the other storage HU. The capacitors and/or inductors of themodel 216 are coupled with each other in a manner, e.g., a parallelmanner, a serial manner, a combination thereof, etc. The model 216represents capacitances and/or inductances of the filters 202 and 208.

Moreover, the system 217 includes the cylinder model 211, which is acomputer-generated model of the RF rod 199 (FIG. 4) and the support 146(FIG. 4). The cylinder model 211 has similar characteristics as that ofelectrical components of the RF rod 199 and the support 146. Thecylinder model 211 includes one or more capacitors, one or moreinductors, connections between the inductors, connections between thecapacitors, and/or connections between a combination of the capacitorsand inductors.

The processor of the host system 130 (FIG. 1) calculates a combinedimpedance, e.g., total impedance, etc., of the model 216, the tunnel andstrap model 210, and the cylinder model 211. The combined impedanceprovides a complex voltage and impedance at the node N2 m. With theinclusion of the model 216 and the tunnel and strap model 210 indetermining the variable at the node N2 m, accuracy of the variable isimproved. It should be noted that an output of the model 216 is themodel node N2 m.

FIG. 8B is a diagram of an embodiment of a system 219 used to illustratea model of the filters 202 and 208 (FIG. 7) to improve an accuracy ofthe variable. The system 219 includes the tunnel and strap model 210 anda model 218, which is coupled in parallel to the tunnel and strap model210. The model 218 is an example of the model 216 (FIG. 8A). The model218 includes an inductor Lfilter, which represents a combined inductanceof the filters 202 and 208. The model 218 further includes a capacitorCfilter, which represents directed combined capacitance of the filters202 and 208.

FIG. 9 is a block diagram of an embodiment of a system 236 for using avoltage and current probe 238 to measure a variable at an output 231 ofan RF generator 220. The output 231 is an example of the node N3(FIG. 1) or of the node N5 (FIG. 1). The RF generator 220 is an exampleof the x MHz generator or the y MHz generator (FIG. 1). The host system130 generates and provides a digital pulsing signal 213 having two ormore states to a digital signal processor (DSP) 226. In one embodiment,the digital pulsing signal 213 is a transistor-transistor logic (TTL)signal. Examples of the states include an on state and an off state, astate having a digital value of 1 and a state having a digital value of0, a high state and a low state, etc.

In another embodiment, instead of the host system 130, a clockoscillator, e.g., a crystal oscillator, etc., is used to generate ananalog clock signal, which is converted by an analog-to-digitalconverter into a digital signal similar to the digital pulsing signal213.

The digital pulsing signal 213 is sent to the DSP 226. The DSP 226receives the digital pulsing signal 213 and identifies the states of thedigital pulsing signal 213. For example, the DSP 226 determines that thedigital pulsing signal 213 has a first magnitude, e.g., the value of 1,the high state magnitude, etc., during a first set of time periods andhas a second magnitude, e.g., the value of 0, the low state magnitude,etc., during a second set of time periods. The DSP 226 determines thatthe digital pulsing signal 213 has a state S1 during the first set oftime periods and has a state S0 during the second set of time periods.Examples of the state S0 include the low state, the state having thevalue of 0, and the off state. Examples of the state S1 include the highstate, the state having the value of 1, and the on state. As yet anotherexample, the DSP 226 compares a magnitude of the digital pulsing signal213 with a pre-stored value to determine that the magnitude of thedigital pulsing signal 213 is greater than the pre-stored value duringthe first set of time periods and that the magnitude during the state S0of the digital pulsing signal 213 is not greater than the pre-storedvalue during the second set of time periods. In the embodiment in whichthe clock oscillator is used, the DSP 226 receives an analog clocksignal from the clock oscillator, converts the analog signal into adigital form, and then identifies the two states S0 and S1.

When a state is identified as S1, the DSP 226 provides a power value P1and/or a frequency value F1 to a parameter control 222. Moreover, whenthe state is identified as S0, the DSP 226 provides a power value P0and/or a frequency value F0 to a parameter control 224. An example of aparameter control that is used to tune a frequency includes an autofrequency tuner (AFT).

It should be noted that the parameter control 222, the parameter control224, and the DSP 226 are portions of a control system 187. For example,the parameter control 222 and the parameter control 224 are logicblocks, e.g., tuning loops, etc., which are portions of a computerprogram that is executed by the DSP 226. In some embodiments, thecomputer program is embodied within a non-transitory computer-readablemedium, e.g., a storage HU.

In an embodiment, a controller, e.g., hardware controller, ASIC, PLD,etc., is used instead of a parameter control. For example, a hardwarecontroller is used instead of the parameter control 222 and anotherhardware controller is used instead of the parameter control 224.

Upon receiving the power value P1 and/or the frequency value F1, theparameter control 222 provides the power value P1 and/or the frequencyvalue F1 to a driver 228 of a drive and amplifier system (DAS) 232.Examples of a driver includes a power driver, a current driver, avoltage driver, a transistor, etc. The driver 228 generates an RF signalhaving the power value P1 and/or the frequency value F1 and provides theRF signal to an amplifier 230 of the DAS 232.

In one embodiment, the driver 228 generates an RF signal having a drivepower value that is a function of the power value P1 and/or having adrive frequency value that is a function of the frequency value F1. Forexample, the drive power value is within a few watts, e.g. 1 thru 5watts, etc., of the power value P1 and the drive frequency value iswithin a few Hz, e.g. 1 thru 5 Hz, etc., of the frequency value F1.

The amplifier 230 amplifies the RF signal having the power value P1and/or the frequency value F1 and generates an RF signal 215 thatcorresponds to the RF signal received from the driver 228. For example,the RF signal 215 has a higher amount of power than that of the powervalue P1. As another example, the RF signal 215 has the same amount ofpower as that of the power value P1. The RF signal 215 is transferredvia a cable 217 and the impedance matching circuit 114 to the ESC 177(FIG. 1).

The cable 217 is an example of the cable 150 or the cable 152 (FIG. 1).For example, when the RF generator 220 is an example of the x MHz RFgenerator (FIG. 1), the cable 217 is an example of the cable 150 andwhen the RF generator 220 is an example of the y MHz RF generator (FIG.1), the cable 217 is an example of the cable 152.

When the power value P1 and/or the frequency value F1 are provided tothe DAS 232 by the parameter control 222 and the RF signal 215 isgenerated, the voltage and current probe 238 measures values of thevariable at the output 231 that is coupled to the cable 217. The voltageand current probe 238 is an example of the voltage and current probe 110or the voltage and current probe 111 (FIG. 1). The voltage and currentprobe 238 sends the values of the variable via a communication device233 to the host system 130 for the host system 130 to execute the method102 (FIG. 3) and methods 340, 351, and 363 (FIGS. 13, 15, and 17)described herein. The communication device 233 is an example of thecommunication device 185 or 189 (FIG. 1). The communication device 233applies a protocol, e.g., Ethernet, EtherCAT, USB, serial, parallel,packetization, depacketization, etc., to transfer data from the voltageand current probe 238 to the host system 130. In various embodiments,the host system 130 includes a communication device that applies theprotocol applied by the communication device 233. For example, when thecommunication 233 applies packetization, the communication device of thehost system 130 applies depacketization. As another example, when thecommunication device 233 applies a serial transfer protocol, thecommunication device of the host system 130 applies a serial transferprotocol.

Similarly, upon receiving the power value P0 and/or the frequency valueF0, the parameter control 224 provides the power value P0 and/or thefrequency value F0 to the driver 228. The driver 228 creates an RFsignal having the power value P0 and/or the frequency value F0 andprovides the RF signal to the amplifier 230.

In one embodiment, the driver 228 generates an RF signal having a drivepower value that is a function of the power value P0 and/or having adrive frequency value that is a function of the frequency value F0. Forexample, the drive power value is within a few, e.g. 1 thru 5, watts ofthe power value P0 and the drive frequency value is within a few, e.g. 1thru 5, Hz of the frequency value F0.

The amplifier 230 amplifies the RF signal having the power value P0and/or the frequency value F0 and generates an RF signal 221 thatcorresponds to the RF signal received from the driver 228. For example,the RF signal 221 has a higher amount of power than that of the powervalue P0. As another example, the RF signal 221 has the same amount ofpower as that of the power value P0. The RF signal 221 is transferredvia the cable 217 and the impedance matching circuit 114 to the knownload 112 (FIG. 2).

When the power value P0 and/or the frequency value F0 are provided tothe DAS 232 by the parameter control 222 and the RF signal 121 isgenerated, the voltage and current probe 238 measures values of thevariable at the output 231. The voltage and current probe 238 sends thevalues of the variable to the host system 130 for the host system 130 toexecute the method 102 (FIG. 2), the method 340 (FIG. 13), the method351 (FIG. 15), or the method 363 (FIG. 17).

It should be noted that the in one embodiment, the voltage and currentprobe 238 is decoupled from the DSP 226. In some embodiments, thevoltage and current probe 238 is coupled to the DSP 226. It shouldfurther be noted that the RF signal 215 generated during the state S1and the RF signal 221 generated during the state S0 are portions of acombined RF signal. For example, the RF signal 215 is a portion of thecombined RF signal that has a higher amount of power than the RF signal221, which is another portion of the combined RF signal.

FIG. 10 is a block diagram of an embodiment of a system 250 in which thevoltage and current probe 238 and the communication device 233 arelocated outside the RF generator 220. In FIG. 1, the voltage and currentprobe 110 is located within the x MHz RF generator to measure thevariable at the output of the x MHz RF generator. The voltage andcurrent probe 238 is located outside the RF generator 220 to measure thevariable at the output 231 of the RF generator 220. The voltage andcurrent probe 238 is associated, e.g., coupled, to the output 231 of theRF generator 220.

FIG. 11 is a block diagram of an embodiment of a system 128 in which thevalues of the variable determined using the system 126 of FIG. 1 areused. The system 128 includes an m MHz RF generator, an n MHz RFgenerator, an impedance matching circuit 115, an RF transmission line287, and a plasma chamber 134. The plasma chamber 134 may be similar tothe plasma chamber 175.

It should be noted that in an embodiment, the x MHz RF generator of FIG.2 is similar to the m MHz RF generator and the y MHz RF generator ofFIG. 2 is similar to the n MHz RF generator. As an example, x MHz isequal to m MHz and y MHz is equal to n MHz. As another example, the xMHz generator and the m MHz generators have similar frequencies and they MHz generator and the n MHz generator have similar frequencies. Anexample of similar frequencies is when the x MHz is within a window,e.g., within kHz or Hz, of the m MHz frequency. In some embodiments, thex MHz RF generator of FIG. 2 is not similar to the m MHz RF generatorand the y MHz RF generator of FIG. 2 is not similar to the n MHz RFgenerator.

It is further noted that in various embodiments, a different type ofsensor is used in each of the m MHz and n MHz RF generators than thatused in each of the x MHz and y MHz RF generators. For example, a sensorthat does not comply with the NIST standard is used in the m MHz RFgenerator. As another example, a voltage sensor that measures onlyvoltage is used in the m MHz RF generator.

It should further be noted that in an embodiment, the impedance matchingcircuit 115 is similar to the impedance matching circuit 114 (FIG. 1).For example, an impedance of the impedance matching circuit 114 is thesame as an impedance of the impedance matching circuit 115. As anotherexample, an impedance of the impedance matching circuit 115 is within awindow, e.g., within 10-20%, of the impedance of the impedance matchingcircuit 114. In some embodiments, the impedance matching circuit 115 isnot similar to the impedance matching circuit 114.

The impedance matching circuit 115 includes electrical components, e.g.,inductors, capacitors, etc., to match an impedance of a power sourcecoupled to the impedance matching circuit 115 with an impedance of aload coupled to the circuit 115. For example, the impedance matchingcircuit 114 matches an impedance of a source coupled to the impedancematching circuit 114, e.g., a combination of the m MHz RF generator, then MHz RF generator, and cables coupling the m and n MHz RF generators tothe impedance matching circuit 114, etc., with an impedance of a load,e.g., a combination of the plasma chamber 134 and the RF transmissionline 287, etc.

It should be noted that in an embodiment, the RF transmission line 287is similar to the RF transmission line 113 (FIG. 1). For example, animpedance of the RF transmission line 287 is the same as an impedance ofthe RF transmission line 113. As another example, an impedance of the RFtransmission line 287 is within a window, e.g., within 10-20%, of theimpedance of the RF transmission line 113. In various embodiments, theRF transmission line 287 is not similar to the RF transmission line 113.

The plasma chamber 134 includes an ESC 192, an upper electrode 264, andother parts (not shown), e.g., an upper dielectric ring surrounding theupper electrode 264, an upper electrode extension surrounding the upperdielectric ring, a lower dielectric ring surrounding a lower electrodeof the ESC 192, a lower electrode extension surrounding the lowerdielectric ring, an upper plasma exclusion zone (PEZ) ring, a lower PEZring, etc. The upper electrode 264 is located opposite to and facing theESC 192. A work piece 262, e.g., a semiconductor wafer, etc., issupported on an upper surface 263 of the ESC 192. Each of the upperelectrode 264 and the lower electrode of the ESC 192 is made of a metal,e.g., aluminum, alloy of aluminum, copper, etc.

In one embodiment, the upper electrode 264 includes a hole that iscoupled to a central gas feed (not shown). The central gas feed receivesone or more process gases from a gas supply (not shown). The upperelectrode 264 is grounded. The ESC 192 is coupled to the m MHz RFgenerator and the n MHz RF generator via the impedance matching circuit115.

When the process gas is supplied between the upper electrode 264 and theESC 192 and when the m MHz RF generator and/or the n MHz RF generatorsupplies power via the impedance matching circuit 115 to the ESC 192,the process gas is ignited to generate plasma within the plasma chamber134.

It should be noted that the system 128 lacks a probe, e.g., a metrologytool, a voltage and current probe, a voltage probe, etc., to measure thevariable at an output 283 of the impedance matching circuit 115, at apoint on the RF transmission line 287, or at the ESC 192. The values ofthe variable at the model nodes N1 m, N2 m, N4 m, and N6 m are used todetermine whether the system 128 is functioning as desired.

In various embodiments, the system 128 lacks a wafer bias sensor, e.g.,an in-situ direct current (DC) probe pick-up pin, and related hardwarethat is used to measure wafer bias at the ESC 192. The nonuse of thewafer bias sensor and the related hardware saves cost.

It should also be noted that in an embodiment, the system 128 includesany number of RF generators coupled to an impedance matching circuit.

FIGS. 12A, 12B, and 12C are diagrams of embodiments of graphs 268, 272,and 275 that illustrate a correlation between voltage, e.g., RMSvoltage, peak voltage, etc., that is measured at the output, e.g., thenode N4, of the impedance matching circuit 114 (FIG. 1) within thesystem 126 (FIG. 1) by using a voltage probe and a voltage, e.g., peakvoltage, etc., at a corresponding model node output, e.g., the node N4m, determined using the method 102 (FIG. 2). Moreover, FIGS. 12A, 12C,and 12C are diagrams of embodiments of graphs 270, 274, and 277 thatillustrate a correlation between current, e.g., root mean square (RMS)current, etc., that is measured the output, e.g., the node N4, of thesystem 126 (FIG. 1) by using a current probe and a current, e.g., RMScurrent, etc., at a corresponding output, e.g., the node N4 m,determined using the method 102 (FIG. 2).

The voltage determined using the method 102 is plotted on an x-axis ineach graph 268, 272, and 275 and the voltage measured with the voltageprobe is plotted on a y-axis in each graph 268, 272, and 275. Similarly,the current determined using the method 102 is plotted on an x-axis ineach graph 270, 274, and 277 and the current measured with the currentprobe is plotted on a y-axis in each graph 270, 274, and 277.

The voltages are plotted in the graph 268 when the x MHz RF generator ison and the y MHz RF generator and a z MHz RF generator, e.g., 60 MHz RFgenerator, are off. Moreover, the voltages are plotted in the graph 272when the y MHz RF generator is on and the x and z MHz RF generators areoff. Also, the voltages are plotted in the graph 275 when the z MHz RFgenerator is on and the x and y MHz RF generators are off.

Similarly, currents are plotted in the graph 270 when the x MHz RFgenerator is on and the y MHz RF generator and a z MHz RF generator areoff. Also, the currents are plotted in the graph 274 when the y MHz RFgenerator is on and the x and z MHz RF generators are off. Also, thecurrents are plotted in the graph 277 when the z MHz RF generator is onand the x and y MHz RF generators are off.

It can be seen in each graph 268, 272, and 275 that an approximatelylinear correlation exists between the voltage plotted on the y-axis ofthe graph and the voltage plotted on the x-axis of the graph. Similarly,it can be seen in each graph 270, 274, and 277 that an approximatelylinear correlation exists between the current plotted on the y-axis andthe current plotted on the x-axis.

FIG. 13 is a flowchart of an embodiment of the method 340 fordetermining a bias at a model node, e.g., the model node N4 m, the modelnode N1 m, the model node N2 m, the model node N6 m, etc., of the plasmasystem 126 (FIG. 1). It should be noted that in some embodiments, waferbias is a direct current (DC) voltage that is created by plasmagenerated within the plasma chamber 175 (FIG. 1). In these embodiments,the wafer bias is present on a surface, e.g., the upper surface 183, ofthe ESC 177 (FIG. 1) and/or on a surface, e.g., an upper surface, of thework piece 131 (FIG. 1).

It should further be noted that the model nodes N1 m and N2 m are on theRF transmission model 161 (FIG. 1) and the model node N6 m is on the ESCmodel 125 (FIG. 1). The method 340 is executed by the processor of thehost system 130 (FIG. 1). In the method 340, the operation 106 isperformed.

Moreover, in an operation 341, one or more models, e.g. the impedancematching model 104, the RF transmission model 161, the ESC model 125(FIG. 1), a combination thereof, etc., of corresponding one or moredevices, e.g., the impedance matching circuit 114, the RF transmissionline 113, the ESC 177, a combination thereof, etc., are generated. Forexample, the ESC model 125 is generated with similar characteristics tothat of the ESC 177 (FIG. 1).

In an operation 343, the complex voltage and current identified in theoperation 106 is propagated through one or more elements of the one ormore models to determine a complex voltage and current at an output ofthe one or more models. For example, the second complex voltage andcurrent is determined from the first complex voltage and current. Asanother example, the second complex voltage and current is determinedfrom the first complex voltage and current and the third complex voltageand current is determined from the second complex voltage and current.As yet another example, the second complex voltage and current isdetermined from the first complex voltage and current, the third complexvoltage and current is determined from the second complex voltage andcurrent, and the third complex voltage and current is propagated throughthe portion 197 of the RF transmission model 161 (FIG. 1) to determine afourth complex voltage and current at the model node N2 m. In thisexample, the fourth complex voltage and current is determined bypropagating the third complex voltage and current through impedances ofelements of the portion 197. As yet another example, the RF transmissionmodel 161 provides an algebraic transfer function that is executed bythe processor of the host system 130 to translate the complex voltageand current measured at one or more outputs of one or more RF generatorsto an electrical node, e.g., the model node N1 m, the model node N2 m,etc., along the RF transmission model 161.

As another example of the operation 343, the second complex voltage andcurrent is determined from the first complex voltage and current, thethird complex voltage and current is determined from the second complexvoltage and current, the fourth complex voltage and current isdetermined from the third complex voltage and current, and the fourthcomplex voltage and current is propagated through the ESC model 125 todetermine a fifth complex voltage and current at the model node N6 m. Inthis example, the fifth complex voltage and current is determined bypropagating the fourth complex voltage and current through impedances ofelements, e.g., capacitors, inductors, etc., of the ESC model 125.

In an operation 342, a wafer bias is determined at the output of the oneor more models based on a voltage magnitude of the complex voltage andcurrent at the output, a current magnitude of the complex voltage andcurrent at the output, and a power magnitude of the complex voltage andcurrent at the output. For example, wafer bias is determined based on avoltage magnitude of the second complex voltage and current, a currentmagnitude of the second complex voltage and current, and a powermagnitude of the second complex voltage and current. To furtherillustrate, when the x MHz RF generator is on and the y MHz and z MHz RFgenerators are off, the processor of the host system 130 (FIG. 1)determines wafer bias at the model node N4 m (FIG. 1) as a sum of afirst product, a second product, a third product, and a constant. Inthis illustration, the first product is a product of a first coefficientand the voltage magnitude of the second complex voltage and current, thesecond product is a product of a second coefficient and the currentmagnitude of the second complex voltage and current, and the thirdproduct is a product of a square root of a third coefficient and asquare root of a power magnitude of the second complex voltage andcurrent.

As an example, a power magnitude is a power magnitude of deliveredpower, which is determined by the processor of the host system 130 as adifference between forward power and reflected power. Forward power ispower supplied by one or more RF generators of the system 126 (FIG. 1)to the plasma chamber 175 (FIG. 1). Reflected power is power reflectedback from the plasma chamber 175 towards one or more RF generators ofthe system 126 (FIG. 1). As an example, a power magnitude of a complexvoltage and current is a determined by the processor of the host system130 as a product of a current magnitude of the complex voltage andcurrent and a voltage magnitude of the complex voltage and current.Moreover, each of a coefficient and a constant used to determine a waferbias is a positive or a negative number. As another example ofdetermination of the wafer bias, when the x MHz RF generator is on andthe y and z MHz RF generators are off, the wafer bias at a model node isrepresented as ax*Vx+bx*Ix+cx*sqrt (Px)+dx, where “ax” is the firstcoefficient, “bx” is the second coefficient, “dx” is the constant, “Vx”is a voltage magnitude of a complex voltage and current at the modelnode “Ix” is a current magnitude of the complex voltage and current atthe model node, and “Px” is a power magnitude of the complex voltage andcurrent at the model node. It should be noted that “sqrt” is a squareroot operation, which is performed by the processor of the host system130. In some embodiments, the power magnitude Px is a product of thecurrent magnitude Ix and the voltage magnitude Vx.

In various embodiments, a coefficient used to determine a wafer bias isdetermined by the processor of the host system 130 (FIG. 1) based on aprojection method. In the projection method, a wafer bias sensor, e.g.,a wafer bias pin, etc., measures wafer bias on a surface, e.g., theupper surface 183 (FIG. 1), etc., of the ESC 177 for a first time.Moreover, in the projection method, a voltage magnitude, a currentmagnitude, and a power magnitude are determined at a model node withinthe plasma system 126 based on complex voltage and current measured atan output of an RF generator. For example, the complex voltage andcurrent measured at the node N3 (FIG. 1) for the first time ispropagated by the processor of the host system 130 to a model node,e.g., the model node N4 m, the model node N1 m, the model node N2 m, orthe model node N6 m (FIG. 1), etc., to determine complex voltage andcurrent at the model node for the first time. Voltage magnitude andcurrent magnitude are extracted by the processor of the host system 130from the complex voltage and current at the model node for the firsttime. Also, power magnitude is calculated by the processor of the hostsystem 130 as a product of the current magnitude and the voltagemagnitude for the first time.

Similarly, in the example, complex voltage and current is measured atthe node N3 for one or more additional times and the measured complexvoltage and current is propagated to determine complex voltage andcurrent at the model node, e.g., the model node N4 m, the model node N1m, the model node N2 m, the model node N6 m, etc., for the one or moreadditional times. Also, for the one or more additional times, voltagemagnitude, current magnitude, and power magnitude are extracted from thecomplex voltage and current determined for the one or more additionaltimes. A mathematical function, e.g., partial least squares, linearregression, etc., is applied by the processor of the host system 130 tothe voltage magnitude, the current magnitude, the power magnitude, andthe measured wafer bias obtained for the first time and for the one ormore additional times to determine the coefficients ax, bx, cx and theconstant dx.

As another example of the operation 342, when the y MHz RF generator ison and the x and z MHz RF generators are off, a wafer bias is determinedas ay*Vy+by*Iy+cy*sqrt (Py)+dy, where “ay” is a coefficient, “by” is acoefficient, “dy” is a constant, “Vy” is a voltage magnitude of thesecond complex voltage and current, “Iy” is a current magnitude of thesecond complex voltage and current, and “Px” is a power magnitude of thesecond complex voltage and current. The power magnitude Py is a productof the current magnitude Iy and the voltage magnitude Vy. As yet anotherexample of the operation 342, when the z MHz RF generator is on and thex and y MHz RF generators are off, a wafer bias is determined asaz*Vz+bz*Iz+cz*sqrt (Pz)+dz, where “az” is a coefficient, “bz” is acoefficient, “dz” is a constant, “Vz” is a voltage magnitude of thesecond complex voltage and current, “Iz” is a current magnitude of thesecond complex voltage and current, and “Pz” is a power magnitude of thesecond complex voltage and current. The power magnitude Pz is a productof the current magnitude Iz and the voltage magnitude Vz.

As another example of the operation 342, when the x and y MHz RFgenerators are on and the z MHz RF generator is off, the wafer bias isdetermined as a sum of a first product, a second product, a thirdproduct, a fourth product, a fifth product, a sixth product, and aconstant. The first product is a product of a first coefficient and thevoltage magnitude Vx, the second product is a product of a secondcoefficient and the current magnitude Ix, the third product is a productof a third coefficient and a square root of the power magnitude Px, thefourth product is a product of a fourth coefficient and the voltagemagnitude Vy, the fifth product is a product of a fifth coefficient andthe current magnitude Iy, and the sixth product is a product of a sixthcoefficient and a square root of the power magnitude Py. When the x andy MHz RF generators are on and the z MHz RF generator is off, the waferbias is represented as axy*Vx+bxy*Ix+cxy*sqrt(Px)+dxy*Vy+exy*Iy+fxy*sqrt (Py)+gxy, where “axy”, “bxy”, “cxy”, “dxy”,“exy”, “fxy”, “dxy”, “exy”, and “fxy” are coefficients, and “gxy” is aconstant.

As another example of the operation 342, when the y and z MHz RFgenerators are on and the x MHz RF generator is off, a wafer bias isdetermined as ayz*Vy+byz*Iy+cyz*sqrt (Py)+dyz*Vz+eyz*Iz+fyz*sqrt(Pz)+gyz, where “ayz”, “byz”, “cyz”, “dyz”, “eyz”, and “fyz” arecoefficients, and “gyz” is a constant. As yet another example of theoperation 342, when the x and z MHz RF generators are on and the y MHzRF generator is off, a wafer bias is determined asaxz*Vx+bxz*Ix+cxz*sqrt (Px)+dxz*Vz+exz*Iz+fxz*sqrt (Pz)+gxz, where“axz”, “bxz”, “cxz”, “dxz”, “exz”, and “fxz” are coefficients, and gxzis a constant.

As another example of the operation 342, when the x, y, and z MHz RFgenerators are on, the wafer bias is determined as a sum of a firstproduct, a second product, a third product, a fourth product, a fifthproduct, a sixth product, a seventh product, an eighth product, a ninthproduct, and a constant. The first product is a product of a firstcoefficient and the voltage magnitude Vx, the second product is aproduct of a second coefficient and the current magnitude Ix, the thirdproduct is a product of a third coefficient and a square root of thepower magnitude Px, the fourth product is a product of a fourthcoefficient and the voltage magnitude Vy, the fifth product is a productof a fifth coefficient and the current magnitude Iy, the sixth productis a product of a sixth coefficient and a square root of the powermagnitude Py, the seventh product is a product of a seventh coefficientand the voltage magnitude Vz, the eighth product is a product of aneighth coefficient and the current magnitude Iz, and the ninth productis a product of a ninth coefficient and a square root of a powermagnitude Pz. When the x, y, and z MHz RF generators are on, the waferbias is represented as axyz*Vx+bxyz*Ix+cxyz*sqrt(Px)+dxyz*Vy+exyz*Iy+fxyz*sqrt (Py)+gxyz*Vz+hxyz*Iz+ixyz*sqrt (Pz)+jxyz,where “axyz”, “bxyz”, “cxyz”, “dxyz”, “exyz”, “fxyz”, “gxyz”, “hxyz”,and “ixyz” are coefficients, and “jxyz” is a constant.

As another example of determination of wafer bias at the output of theone or more models, wafer bias at the model node N1 m is determined bythe processor of the host system 130 based on voltage and currentmagnitudes determined at the model node N1 m. To further illustrate, thesecond complex voltage and current is propagated along the portion 173(FIG. 1) to determine complex voltage and current at the model node N1m. The complex voltage and current is determined at the model node N1 mfrom the second complex voltage and current in a manner similar to thatof determining the second complex voltage and current from the firstcomplex voltage and current. For example, the second complex voltage andcurrent is propagated along the portion 173 based on characteristics ofelements of the portion 173 to determine a complex voltage and currentat the model node N1 m.

Based on the complex voltage and current determined at the model node N1m, wafer bias is determined at the model node N1 m by the processor ofthe host system 130. For example, wafer bias is determined at the modelnode N1 m from the complex voltage and current at the model node N1 m ina manner similar to that of determining the wafer bias at the model nodeN4 m from the second complex voltage and current. To illustrate, whenthe x MHz RF generator is on and the y MHz and z MHz RF generators areoff, the processor of the host system 130 (FIG. 1) determines wafer biasat the model node N1 m as a sum of a first product, a second product, athird product, and a constant. In this example, the first product is aproduct of a first coefficient and the voltage magnitude of the complexvoltage and current at the model node N1 m, the second product is aproduct of a second coefficient and the current magnitude of the complexvoltage and current at the model node N1 m, and the third product is aproduct of a square root of a third coefficient and a square root of apower magnitude of the complex voltage and current at the model node N1m. When the x MHz RF generator is on and the y and z MHz RF generatorsare off, the wafer bias at the model node N1 m is represented asax*Vx+bx*Ix+cx*sqrt (Px)+dx, where ax is the first coefficient, bx isthe second coefficient, cx is the third coefficient, dx is the constant,Vx is the voltage magnitude at the model node N1 m, Ix is the currentmagnitude at the model node N1 m, and Px is the power magnitude at themodel node N1 m.

Similarly, based on the complex voltage and current at the model node N1m and based on which of the x, y, and z MHz RF generators are on, thewafer bias ay*Vy+by*Iy+cy*sqrt (Py)+dy, az*Vz+bz*Iz+cz*sqrt (Pz)+dz,axy*Vx+bxy*Ix+cxy*sqrt (Px)+dxy*Vy+exy*Iy+fxy*sqrt (Py)+gxy,axz*Vx+bxz*Ix+cxz*sqrt (Px)+dxz*Vz+exz*Iz+fxz*sqrt (Pz)+gxz,ayz*Vy+byz*Iy+cyz*sqrt (Py)+dyz*Vz+eyz*Iz+fyz*sqrt (Pz)+gyz, andaxyz*Vx+bxyz*Ix+cxyz*sqrt (Px)+dxyz*Vy+exyz*Iy+fxyz*sqrt(Py)+gxyz*Vz+hxyz*Iz+ixyz*sqrt (Pz)+jxyz are determined.

As yet another example of determination of wafer bias at the output ofthe one or more models, wafer bias at the model node N2 m is determinedby the processor of the host system 130 based on voltage and currentmagnitudes determined at the model node N2 m in a manner similar to thatof determining wafer bias at the model node N1 m based on voltage andcurrent magnitudes determined at the model node N1 m. To furtherillustrate, wafer bias ax*Vx+bx*Ix+cx*sqrt (Px)+dx, ay*Vy+by*Iy+cy*sqrt(Py)+dy, az*Vz+bz*Iz+cz*sqrt (Pz)+dz, axy*Vx+bxy*Ix+cxy*sqrt(Px)+dxy*Vy+exy*Iy+fxy*sqrt (Py)+gxy, axz*Vx+bxz*Ix+cxz*sqrt(Px)+dxz*Vz+exz*Iz+fxz*sqrt (Pz)+gxz, ayz*Vy+byz*Iy+cyz*sqrt(Py)+dyz*Vz+eyz*Iz+fyz*sqrt (Pz)+gyz, and axyz*Vx+bxyz*Ix+cxyz*sqrt(Px)+dxyz*Vy+exyz*Iy+fxyz*sqrt (Py)+gxyz*Vz+hxyz*Iz+ixyz*sqrt (Pz)+jxyzare determined at the model node N2 m.

As another example of determination of wafer bias at the output of theone or more models, wafer bias at the model node N6 m is determined bythe processor of the host system 130 based on voltage and currentmagnitudes determined at the model node N6 m in a manner similar to thatof determining wafer bias at the model node N2 m based on voltage andcurrent magnitudes determined at the model node N2 m. To furtherillustrate, wafer bias ax*Vx+bx*Ix+cx*sqrt (Px)+dx, ay*Vy+by*Iy+cy*sqrt(Py)+dy, az*Vz+bz*Iz+cz*sqrt (Pz)+dz, axy*Vx+bxy*Ix+cxy*sqrt(Px)+dxy*Vy+exy*Iy+fxy*sqrt (Py)+gxy, axz*Vx+bxz*Ix+cxz*sqrt(Px)+dxz*Vz+exz*Iz+fxz*sqrt (Pz)+gxz, ayz*Vy+byz*Iy+cyz*sqrt(Py)+dyz*Vz+eyz*Iz+fyz*sqrt (Pz)+gyz, and axyz*Vx+bxyz*Ix+cxyz*sqrt(Px)+dxyz*Vy+exyz*Iy+fxyz*sqrt (Py)+gxyz*Vz+hxyz*Iz+ixyz*sqrt (Pz)+jxyzare determined at the model node N6 m.

It should be noted that in some embodiments, wafer bias is stored withinthe storage HU 162 (FIG. 1).

FIG. 14 is a state diagram illustrating an embodiment of a wafer biasgenerator 340, which is implemented within the host system 130 (FIG. 1).When all of the x, y, and z MHz RF generators are off, wafer bias iszero or minimal at a model node, e.g., the model node N4 m, N1 m, N2 m,N6 m (FIG. 1), etc. When the x, y, or z MHz RF generator is on and theremaining of the x, y, and z MHz RF generators are off, the wafer biasgenerator 340 determines a wafer bias at a model node, e.g., the modelnode N4 m, N1 m, N2 m, N6 m, etc., as a sum of a first product a*V, asecond product b*I, a third product c*sqrt(P), and a constant d, where Vis a voltage magnitude of a complex voltage and current at the modelnode, I is a current magnitude of the complex voltage and current, P isa power magnitude of the complex voltage and current, a is acoefficient, b is a coefficient, c is a coefficient, and d is aconstant. In various embodiments, a power magnitude at a model node is aproduct of a current magnitude at the model node and a voltage magnitudeat the model node. In some embodiments, the power magnitude is amagnitude of delivered power.

When two of the x, y, and z MHz RF generators are on and the remainingof the x, y, and z MHz RF generators are off, the wafer bias generator340 determines a wafer bias at a model node, e.g., the model node N4 m,N1 m, N2 m, N6 m, etc., as a sum of a first product a12*V1, a secondproduct b12*I1, a third product c12*sqrt(P1), a fourth product d12*V2, afifth product e12*I2, a sixth product f12*sqrt(P2), and a constant g12,where “V1” is a voltage magnitude of a complex voltage and current atthe model node determined by propagating a voltage measured at an outputof a first one of the RF generators that is on, “H” is a currentmagnitude of the complex voltage and current determined by propagating acurrent measured at the output of the first RF generator that is on,“P1” is a power magnitude of the complex voltage and current determinedas a product of V1 and I1, “V2” is a voltage magnitude of the complexvoltage and current at the model node determined by propagating avoltage measured at an output of a second one of the RF generators thatis on, I2″ is a current magnitude of the complex voltage and currentdetermined by propagating the current measured at an output of thesecond RF generator that is on, “P2” is a power magnitude determined asa product of V2 and I2, each of “a12”, “b12”, “c12”, “d12”, “e12” and“f12” is a coefficient, and “g12” is a constant.

When all of the x, y, and z MHz RF generators are on, the wafer biasgenerator 340 determines a wafer bias at a model node, e.g., the modelnode N4 m, N1 m, N2 m, N6 m, etc., as a sum of a first product a123*V1,a second product b123*I1, a third product c123*sqrt(P1), a fourthproduct d123*V2, a fifth product e123*I2, a sixth product f123*sqrt(P2),a seventh product g123*V3, an eighth product h123*I3, a ninth producti123*sqrt(P3), and a constant j123, where “V1” is a voltage magnitude ofa complex voltage and current at the model node determined bypropagating a voltage measured at an output of a first one of the RFgenerators, “H” is a current magnitude of the complex voltage andcurrent determined by propagating a current measured at the output ofthe first RF generator, “P1” is a power magnitude of the complex voltageand current determined as a product of V1 and I1, “V2” is a voltagemagnitude of the complex voltage and current at the model nodedetermined by propagating a voltage measured at an output of a secondone of the RF generators, I2″ is a current magnitude of the complexvoltage and current determined by propagating a current measured at theoutput of the second RF generator, “P2” is a power magnitude of thecomplex voltage and current determined as a product of V2 and I2, “V3”is a voltage magnitude of the complex voltage and current at the modelnode determined by propagating a voltage measured at an output of athird one of the RF generators, “I3” is a current magnitude of thecomplex voltage and current determined by propagating a current at theoutput of the third RF generator, “P3” is a power magnitude of thecomplex voltage and current determined as a product of V3 and I3, eachof “a123”, “b123”, “c123”, “d123”, “e123”, “f123”, “g123”, “h123”, and“i123” is a coefficient, and “j123” is a constant.

FIG. 15 is a flowchart of an embodiment of the method 351 fordetermining a wafer bias at a point along a path 353 (FIG. 16) betweenthe model node N4 m (FIG. 16) and the ESC model 125 (FIG. 16). FIG. 15is described with reference to FIG. 16, which is a block diagram of anembodiment of a system 355 for determining a wafer bias at an output ofa model.

In an operation 357, output of the x, y, or z MHz RF generator isdetected to identify a generator output complex voltage and current. Forexample, the voltage and current probe 110 (FIG. 1) measures complexvoltage and current at the node N3 (FIG. 1). In this example, thecomplex voltage and current is received from the voltage and currentprobe 110 via the communication device 185 (FIG. 1) by the host system130 (FIG. 1) for storage within the storage HU 162 (FIG. 1). Also, inthe example, the processor of the host system 130 identifies the complexvoltage and current from the storage HU 162.

In an operation 359, the processor of the host system 130 uses thegenerator output complex voltage and current to determine a projectedcomplex voltage and current at a point along the path 353 between themodel node N4 m and the model node N6 m. The path 161 extends from themodel node N4 m to the model node N6 m. For example, the fifth complexvoltage and current is determined from the complex voltage and currentmeasured at the output of the x MHz RF generator, the y MHz RFgenerator, or the z MHz RF generator. As another example, the complexvoltage and current measured at the node N3 or the node N5 is propagatedvia the impedance matching model 104 to determine a complex voltage andcurrent at the model node N4 m (FIG. 1). In the example, the complexvoltage and current at the model node N4 m is propagated via one or moreelements of the RF transmission model 161 (FIG. 16) and/or via one ormore elements of the ESC model 125 (FIG. 16) to determine complexvoltage and current at a point on the path 353.

In an operation 361, the processor of the host system 130 applies theprojected complex voltage and current determined at the point on thepath 353 as an input to a function to map the projected complex voltageand current to a wafer bias value at the node N6 m of the ESC model 125(FIG. 15). For example, when the x, y, or z MHz RF generator is on, awafer bias at the model node N6 m is determined as a sum of a firstproduct a*V, a second product b*I, a third product c*sqrt(P), and aconstant d, where, V is a voltage magnitude of the projected complexvoltage and current at the model node N6 m, I is a current magnitude ofthe projected complex voltage and current at the model node N6 m, P is apower magnitude of the projected complex voltage and current at themodel node N6 m, a, b, and c are coefficients, and d is a constant.

As another example, when two of the x, y, and z MHz RF generators are onand the remaining of the x, y, and z MHz RF generators are off, a waferbias at the model node N6 m is determined as a sum of a first producta12*V1, a second product b12*I1, a third product c12*sqrt(P1), a fourthproduct d12*V2, a fifth product e12*I2, a sixth product f12*sqrt(P2),and a constant g12, where V1 is a voltage magnitude at the model node N6m as a result of a first one of the two RF generators being on, I1 is acurrent magnitude at the model node N6 m as a result of the first RFgenerator being on, P1 is a power magnitude at the model node N6 m as aresult of the first RF generator being on, V2 is a voltage magnitude atthe model node N6 m as a result of a second one of the two RF generatorsbeing on, I2 is a current magnitude at the model node N6 m as a resultof the second RF generator being on, and P2 is a power magnitude at themodel node N6 m as a result of the second RF generator being on, a12,b12, c12, d12, e12, and f12 are coefficients, and g12 is a constant.

As yet another example, when all of the x, y, and z MHz RF generatorsare on, a wafer bias at the model node N6 m is determined as a sum of afirst product a123*V1, a second product b123*I1, a third productc123*sqrt(P1), a fourth product d123*V2, a fifth product e123*I2, asixth product f123*sqrt(P2), a seventh product g123*V3, an eighthproduct h123*I3, a ninth product i123*sqrt(P3), and a constant j123,where V1, I1, P1, V2, I2, and P2 are described above in the precedingexample, V3 is a voltage magnitude at the model node N6 m as a result ofa third one of the RF generators being on, I3 is a current magnitude atthe model node N6 m as a result of the third RF generator being on, andP3 is a power magnitude at the model node N6 m as a result of the thirdRF generator being on, a123, b123, c123, d123, e123, f123, g123, h123,and i123 are coefficients and j123 is a constant.

As another example, a function used to determine a wafer bias is a sumof characterized values and a constant. The characterized values includemagnitudes, e.g., the magnitudes V, I, P, V1, I1, P1, V2, I2, P2, V3,I3, P3, etc. The characterized values also include coefficients, e.g.,the coefficients, a, b, c, a12, b12, c12, d12, e12, f12, a123, b123,c123, d123, e123, f123, g123, h123, i123, etc. Examples of the constantinclude the constant d, the constant g12, the constant j123, etc.

It should be noted that the coefficients of the characterized values andthe constant of the characterized values incorporate empirical modelingdata. For example, wafer bias is measured for multiple times at the ESC177 (FIG. 1) using a wafer bias sensor. Moreover, in the example, forthe number of times the wafer bias is measured, complex voltages andcurrents at the point along the path 353 (FIG. 16) are determined bypropagating the complex voltage and current from one or more of thenodes, e.g., the nodes N3, N5, etc., of one or more of the RFgenerators, e.g., the x MHz RF generator, the y MHz RF generator, the zMHz RF generator, etc., via one or more of the models, e.g., theimpedance matching model 104, the model portion 173, the RF transmissionmodel 161, the ESC model 125 (FIG. 1), to reach to the point on the path353 (FIG. 16). Moreover, in this example, a statistical method, e.g.,partial least squares, regression, etc., is applied by the processor ofthe host system 130 to the measured wafer bias and to voltagemagnitudes, current magnitudes, and power magnitudes extracted from thecomplex voltages and currents at the point to determine the coefficientsof the characterized values and the constant of the characterizedvalues.

In various embodiments, a function used to determine a wafer bias ischaracterized by a summation of values that represent physicalattributes of the path 353. The physical attributes of the path 353 arederived values from test data, e.g., empirical modeling data, etc.Examples of physical attributes of the path 353 include capacitances,inductances, a combination thereof, etc., of elements on the path 353.As described above, the capacitances and/or inductances of elementsalong the path 353 affect voltages and currents empirically determinedusing the projection method at the point on the path 353 and in turn,affect the coefficients of the characterized values and the constant ofthe characterized values.

In some embodiments, a function used to determine a wafer bias is apolynomial.

FIG. 17 is a flowchart of an embodiment of the method 363 fordetermining a wafer bias at a model node of the system 126 (FIG. 1).FIG. 17 is described with reference to FIGS. 1 and 16. The method 363 isexecuted by the processor of the host system 130 (FIG. 1). In anoperation 365, one or more complex voltages and currents are received bythe host system 130 from one or more communication devices of agenerator system, which includes one or more of the x MHz RF generator,the y MHz RF generator, and the z MHz RF generator. For example, complexvoltage and current measured at the node N3 is received from thecommunication device 185 (FIG. 1). As another example, complex voltageand current measured at the node N5 is received from the communicationdevice 189 (FIG. 1). As yet another example, complex voltage and currentmeasured at the node N3 and complex voltage and current measured at thenode N5 are received. It should be noted that an output of the generatorsystem includes one or more of the nodes N3, N5, and an output node ofthe z MHz RF generator.

In an operation 367, based on the one or more complex voltages andcurrents at the output of the generator system, a projected complexvoltage and current is determined at a point along, e.g., on, etc., thepath 353 (FIG. 16) between the impedance matching model 104 and the ESCmodel 125 (FIG. 16). For example, the complex voltage and current at theoutput of the generator system is projected via the impedance matchingmodel 104 (FIG. 16) to determine a complex voltage and current at themodel node N4 m. As another example, the complex voltage and current atthe output of the generator system is projected via the impedancematching model 104 and the portion 173 (FIG. 1) of the RF transmissionmodel 161 to determine a complex voltage and current at the model nodeN1 m (FIG. 1). As yet another example, the complex voltage and currentat the output of the generator system is projected via the impedancematching model 104 and the RF transmission model 161 to determine acomplex voltage and current at the model node N2 m (FIG. 1). As anotherexample, the complex voltage and current at the output of the generatorsystem is projected via the impedance matching model 104, the RFtransmission model 161, and the ESC model 125 to determine a complexvoltage and current at the model node N6 m (FIG. 1).

In an operation 369, a wafer bias is calculated at the point along thepath 353 by using the projected complex V&I as an input to a function.For example, when the x, y, or z MHz RF generator is on and theremaining of the x, y, and z MHz RF generators are off, a wafer bias atthe point is determined from a function, which is as a sum of a firstproduct a*V, a second product b*I, a third product c*sqrt(P), and aconstant d, where, V is a voltage magnitude of the projected complexvoltage and current at the point, I is a current magnitude of theprojected complex voltage and current at the point, P is a powermagnitude of the projected complex voltage and current at the point, a,b, and c are coefficients, and d is a constant.

As another example, when two of the x, y, and z MHz RF generators are onand the remaining of the x, y, and z MHz RF generators are off, a waferbias at the point is determined as a sum of a first product a12*V1, asecond product b12*I1, a third product c12*sqrt(P1), a fourth productd12*V2, a fifth product e12*I2, a sixth product f12*sqrt(P2), and aconstant g12, where V1 is a voltage magnitude at the point as a resultof a first one of the two RF generators being on, I1 is a currentmagnitude at the point as a result of the first RF generator being on,P1 is a power magnitude at the point as a result of the first RFgenerator being on, V2 is a voltage magnitude at the point as a resultof a second one of the two RF generators being on, I2 is a currentmagnitude at the point as a result of the second RF generator being on,and P2 is a power magnitude at the point as a result of the second RFgenerator being on, a12, b12, c12, d12, e12, and f12 are coefficients,and g12 is a constant.

As yet another example, when all of the x, y, and z MHz RF generatorsare on, a wafer bias at the point is determined as a sum of a firstproduct a123*V1, a second product b123*I1, a third productc123*sqrt(P1), a fourth product d123*V2, a fifth product e123*I2, asixth product f123*sqrt(P2), a seventh product g123*V3, an eighthproduct h123*I3, a ninth product i123*sqrt(P3), and a constant j123,where V1, I1, P1, V2, I2, and P2 are described above in the precedingexample, V3 is a voltage magnitude at the point as a result of a thirdone of the RF generators being on, I3 is a current magnitude at thepoint as a result of the third RF generator being on, and P3 is a powermagnitude at the point as a result of the third RF generator being on,a123, b123, c123, d123, e123, f123, g123, h123, and i123 arecoefficients, and j123 is a constant.

FIG. 18 is a block diagram of an embodiment of a system 330 that is usedto illustrate advantages of determining wafer bias by using the method340 (FIG. 13), the method 351 (FIG. 15), or the method 363 (FIG. 17)instead of by using a voltage probe 332, e.g., a voltage sensor, etc.

The voltage probe 332 is coupled to the node N1 to determine a voltageat the node N1. In some embodiments, the voltage probe 332 is coupled toanother node, e.g., node N2, N4, etc., to determine voltage at the othernode. The voltage probe 332 includes multiple circuits, e.g., an RFsplitter circuit, a filter circuit 1, a filter circuit 2, a filtercircuit 3, etc.

Also, the x and y MHz RF generators are coupled to a host system 334that includes a noise or signal determination module 336. It should benoted that a module may be a processor, an ASIC, a PLD, a softwareexecuted by a processor, or a combination thereof.

The voltage probe 332 measures a voltage magnitude, which is used by thehost system 334 to determine a wafer bias. The module 336 determineswhether the voltage magnitude measured by the voltage probe 332 is asignal or noise. Upon determining that the voltage magnitude measured bythe voltage probe 332 is a signal, the host system 334 determines waferbias.

The system 126 (FIG. 1) is cost effective compared to the system 330 andsaves time and effort compared to the system 330. The system 330includes the voltage probe 332, which does not need to be included inthe system 126. There is no need to couple a voltage probe at the nodeN4, N1, or N2 of the system 126 to determine wafer bias. In the system126, wafer bias is determined based on the impedance matching model 104,RF transmission model 161, and/or the ESC model 125 (FIG. 1). Moreover,the system 330 includes the module 336, which also does not need to beincluded in the system 126. There is no need to spend time and effort todetermine whether a complex voltage and current is a signal or noise. Nosuch determination needs to be made by the host system 130 (FIG. 1).

FIGS. 19A, 19B, and 19C show embodiments of graphs 328, 332, and 336 toillustrate a correlation, e.g., a linear correlation, etc., betweenvoltage, e.g., peak voltage, etc., that is measured at the output, e.g.,the node N1, of the portion 195 (FIG. 1) by using a voltage probe and avoltage, e.g., peak voltage, etc., at a corresponding model node output,e.g., the node N1 m, determined using the method 102 (FIG. 2). In eachgraph 328, 332, and 336, the measured voltage is plotted on a y-axis andthe voltage determined using the method 102 is plotted on an x-axis.

Moreover, FIGS. 19A, 19B, and 19C show embodiments of graphs 330, 334,and 338 to illustrate a correlation, e.g., a linear correlation, etc.,between wafer bias that is measured at the output N6 (FIG. 1) by using awafer bias probe and wafer bias at a corresponding model node output,e.g., the node N6 m, determined using the method 340 (FIG. 13), themethod 351 (FIG. 15), or the method 363 (FIG. 17). In each graph 330,334, and 338, the wafer bias determined using the wafer bias probe isplotted on a y-axis and the wafer bias determined using the method 340,the method 351, or the method 363 is plotted on an x-axis.

The voltages and wafer bias are plotted in the graphs 328 and 330 whenthe y MHz and z MHz RF generators are on and the x MHz RF generator isoff. Moreover, the voltages and wafer bias are plotted in the graphs 332and 334 when the x MHz and z MHz RF generators are on and the y MHz RFgenerator is off. Also, the voltages are plotted in the graphs 336 and338 when the x MHz and y MHz RF generators are on and the z MHz RFgenerator is off.

FIG. 20A is a diagram of an embodiment of graphs 276 and 278 toillustrate that there is a correlation between a wired wafer biasmeasured using a sensor tool, e.g., a metrology tool, a probe, a sensor,a wafer bias probe, etc., a model wafer bias that is determined usingthe method 340 (FIG. 13), the method 351 (FIG. 15), or the method 363(FIG. 17), and an error in the model bias. The wired wafer bias that isplotted in the graph 276 is measured at a point, e.g., a node on the RFtransmission line 113, a node on the upper surface 183 (FIG. 1) of theESC 177, etc. and the model bias that is plotted in the graph 276 isdetermined at the corresponding model point, e.g., the model node N4 m,the model node N1 m, the model node N2 m, the model node N6 m, etc.(FIG. 1), on the path 353 (FIG. 16). The wired wafer bias is plottedalong a y-axis in the graph 276 and the model bias is plotted along anx-axis in the graph 276.

The wired wafer bias and the model bias are plotted in the graph 276when the x MHz RF generator is on, and the y and z MHz RF generators areoff. Moreover, the model bias of graph 276 is determined using anequation a2*V2+b2*I2+c2*sqrt (P2)+d2, where “*” representsmultiplication, “sqrt” represents a square root, “V2” represents voltageat the point along the path 353 (FIG. 16), I2 represents current at thepoint, P2 represents power at the point, “a2” is a coefficient, “b2” isa coefficient, “c2” is a coefficient, and “d2” is a constant value.

The graph 278 plots an error, which is an error in the model bias at thepoint, on a y-axis and plots the model bias at the point on an x-axis.The model error is an error, e.g., a variance, a standard deviation,etc., in the model bias. The model error and the model bias are plottedin the graph 278 when the x MHz RF generator is on and the y and z MHzRF generators are off.

FIG. 20B is a diagram of an embodiment of graphs 280 and 282 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 280 and 282 are plotted in a manner similar to the graphs 276and 278 (FIG. 20A) except that the graphs 280 and 282 are plotted whenthe y MHz RF generator is on and the x and z MHz RF generators are off.Moreover, the model bias of the graphs 280 and 282 is determined usingan equation a27*V27+b27*I27+c27*sqrt (P27)+d27, where “V27” represents avoltage magnitude at the point along the path 353 (FIG. 16), “I27”represents a current magnitude at the point, “P27” represents a powermagnitude at the point, “a27” is a coefficient, “b27” is a coefficient,“c27” is a coefficient, and “d27” is a constant value.

FIG. 20C is a diagram of an embodiment of graphs 284 and 286 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 284 and 286 are plotted in a manner similar to the graphs 276and 278 (FIG. 20A) except that the graphs 284 and 286 are plotted whenthe z MHz RF generator is on and the x and y MHz RF generators are off.Moreover, the model bias of the graphs 284 and 286 is determined usingan equation a60*V60+b60*I60+c60*sqrt (P60)+d60, where “V60” represents avoltage magnitude at the point along the path 353 (FIG. 16), I60″represents a current magnitude at the point, “P60” represents a powermagnitude at the point, “a60” is a coefficient, “b60” is a coefficient,“c60” is a coefficient, and “d60” is a constant value.

FIG. 20D is a diagram of an embodiment of graphs 288 and 290 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 288 and 290 are plotted in a manner similar to the graphs 276and 278 (FIG. 20A) except that the graphs 288 and 290 are plotted whenthe x and y MHz RF generators are on, and the z MHz RF generator is off.Moreover, the model bias of the graphs 288 and 290 is determined usingan equation a227*V2+b227*I2+c227*sqrt (P2)+d227*V27+e227*I27+f227*sqrt(P27)+g227, where “a227”, “b227” and “c227”, “d227”, “e227” and “f227”are coefficients, and “g227” is a constant value.

FIG. 20E is a diagram of an embodiment of graphs 292 and 294 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 292 and 294 are plotted in a manner similar to the graphs 276and 278 (FIG. 20A) except that the graphs 292 and 294 are plotted whenthe x and z MHz RF generators are on, and the y MHz RF generator is off.Moreover, the model bias of the graphs 292 and 294 is determined usingan equation a260*V2+b260*I2+c260*sqrt (P2)+d20*V60+e260*I60+f260*sqrt(P60)+g260, where “a260”, “b260” “c260”, “d260”, “e260” and “f260” arecoefficients, and “g260” is a constant value.

FIG. 20F is a diagram of an embodiment of graphs 296 and 298 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 296 and 298 are plotted in a manner similar to the graphs 276and 278 (FIG. 20A) except that the graphs 296 and 298 are plotted whenthe y and z MHz RF generators are on, and the x MHz RF generator is off.Moreover, the model bias of the graphs 296 and 298 is determined usingan equation a2760*V27+b2760*I27+c2760*sqrt(P27)+d2760*V60+e2760*I60+f2760*sqrt (P60)+g2760, where “a2760”, “b2760”“c2760”, “d2760”, “e2760” and “f2760” are coefficients, and “g2760” is aconstant value.

FIG. 20G is a diagram of an embodiment of graphs 302 and 304 toillustrate that there is a correlation between a wired wafer bias, amodel bias that is determined using the method 340 (FIG. 13), the method351 (FIG. 15) or method 363 (FIG. 17), and an error in the model bias.The graphs 302 and 304 are plotted in a manner similar to the graphs 276and 278 (FIG. 20A) except that the graphs 302 and 304 are plotted whenthe x, y and z MHz RF generators are on. Moreover, the model bias of thegraphs 302 and 304 is determined using an equationa22760*V2+b22760*I2+c22760*sqrt (P2)+d22760*V60+e22760*I60+f22760*sqrt(P60)+g22760*V27+h22760*I27+i22760*sqrt (P27)+j22760, where “a22760”,“b22760”, “c22760”, “d22760”, “e22760”, “f22760” “g22760”, “h22760”, and“i22760” are coefficients and “j22760” is a constant value.

FIG. 21 is a block diagram of an embodiment of the host system 130. Thehost system 130 includes a processor 168, the storage HU 162, an inputHU 380, an output HU 382, an input/output (I/O) interface 384, an I/Ointerface 386, a network interface controller (NIC) 388, and a bus 390.The processor 168, the storage HU 162, the input HU 380, the output HU382, the I/O interface 384, the I/O interface 386, and the NIC 388 arecoupled with each other via a bus 392. Examples of the input HU 380include a mouse, a keyboard, a stylus, etc. Examples of the output HU382 include a display, a speaker, or a combination thereof. The displaymay be a liquid crystal display, a light emitting diode display, acathode ray tube, a plasma display, etc. Examples of the NIC 388 includea network interface card, a network adapter, etc.

Examples of an I/O interface include an interface that providescompatibility between pieces of hardware coupled to the interface. Forexample, the I/O interface 384 converts a signal received from the inputHU 380 into a form, amplitude, and/or speed compatible with the bus 392.As another example, the I/O interface 386 converts a signal receivedfrom the bus 392 into a form, amplitude, and/or speed compatible withthe output HU 382.

It should be noted that in some embodiments, wafer bias is used todetermine a clamping voltage that is used to clamp the work piece 131(FIG. 1) to the ESC 177 (FIG. 1). For example, when wafer bias is absentfrom the plasma chamber 175 (FIG. 1), two electrodes within the ESC 177have matching voltages with opposite polarities to clamp the work piece131 to the ESC 177. In the example, when the wafer bias is presentwithin the plasma chamber 175, voltages supplied to the two electrodesare different in magnitude to compensate for the presence of the waferbias. In various embodiments, wafer bias is used to compensate for biasat the ESC 177 (FIG. 1).

It is also noted that the use of three parameters, e.g., currentmagnitude, voltage magnitude, and phase between the current and voltage,etc., to determine wafer bias compared to use of voltage to compensatefor bias at the ESC 177 allows better determination of wafer bias. Forexample, wafer bias calculated using the three parameters has a strongercorrelation to non-linear plasma regimes compared to a relation betweenRF voltage and the non-linear plasma regimes. As another example, waferbias calculated using the three parameters is more accurate than thatdetermined using a voltage probe.

FIG. 22 is a diagram of an embodiment of a function used to illustratedetermination of ion energy from a wafer bias and peak magnitude. Thedetermination of ion energy is performed by the processor 168 (FIG. 21)of the host system 130. For example, the ion energy is calculated as asum of a coefficient “C1” multiplied by a wafer bias, e.g., modeledbias, etc., at the model node N6 m and a coefficient “C2” multiplied bya peak magnitude of a voltage of one or more RF generators. Examples ofthe coefficient “C1” include a negative real number and of thecoefficient “C2” include a positive real number.

In various embodiments, the coefficient “C1” is a positive real number.In various embodiments, the coefficient “C2” is a negative real number.The coefficients “C1” and “C2”, the wafer bias, and the peak magnitudeare stored in the storage HU 162 (FIG. 21). Examples of the peakmagnitude include a peak-to-peak magnitude and a zero-to-peak magnitude.

In some embodiments, the peak magnitude used to determine the ion energyis extracted by the processor 168 of the host system 130 from thecomplex voltage and current at the model node N6 m (FIG. 1). In variousembodiments, the peak magnitude used to determine the ion energy isextracted by the processor 168 of the host system 130 from the complexvoltage and current at the model node N2 m, or the model node N1 m, orthe model node N4 m (FIG. 1).

In various embodiments, the peak magnitude used to calculate the ionenergy is measured by a voltage and current probe that is coupled to thenode N1, the node N2 (FIG. 1), or the node N6 (FIG. 1) at one end and tothe processor 168 at another end. The voltage and current probe coupledto the node N1, the node N2, or the node N6 is capable of distinguishingbetween frequencies of the x and y MHz RF generator.

In some embodiments, both the peak magnitude and wafer bias used todetermine the ion energy is at a model node. For example, the peakmagnitude used to determine the ion energy is extracted from complexvoltage and current at the model node N6 m, and the wafer bias used todetermine the ion energy is calculated at the model node N6 m. Asanother example, the peak magnitude used to determine the ion energy isextracted from complex voltage and current at the model node N2 m, andthe wafer bias used to determine the ion energy is calculated at themodel node N2 m.

In a variety of embodiments, the peak magnitude used to determine theion energy is extracted from a complex voltage and current at a firstmodel node and wafer bias used to determine the ion energy is determinedat a second model node, other than the first model node. For example,the peak magnitude used to determine the ion energy is extracted fromcomplex voltage and current at the model node N6 m, and the wafer biasused to determine the ion energy is calculated at the model node N2 m.As another example, the peak magnitude used to determine the ion energyis extracted from complex voltage and current at the model node N2 m,and the wafer bias used to determine the ion energy is calculated at themodel node N6 m.

In several embodiments, the peak magnitude used to calculate the ionenergy is a voltage at one or more outputs, e.g., the node N3, the nodeN5, etc. (FIG. 1) of one or more of the x and y MHz RF generators (FIG.1). In embodiments in which multiple RF generators are used, e.g., boththe x and y MHz RF generators are used, a peak voltage is measured by avoltage and current probe coupled to the node N3 and to the processor168 and a peak voltage is measured by a voltage and current probecoupled to the node N5 and to the processor 168, and the processor 168calculates an algebraic combination, e.g., a sum, a mean, etc., of thepeak voltages measured at the outputs to calculate the peak magnitudethat is used to calculate the ion energy. An example of a voltage andcurrent probe that is coupled to any of the nodes N3 and N5 includes aNIST probe.

In some embodiments, instead of the peak magnitude, a root mean squaremagnitude is used.

In various embodiments, ion energy is determined by the processor 168 ofthe host system 130 as a function of the wafer bias and an RF voltage,e.g., Vx, Vy, Vz, etc., used to calculate the wafer bias. For example,the processor of the host system 130 determines the ion energy as:Ei=(−½)Vdc+(½)Vpeak

where Ei is the ion energy, Vdc is the wafer bias potential and Vpeak isa zero-to-peak voltage that is used to calculate the wafer biaspotential. The Vpeak is a peak voltage, e.g., the voltage Vx, Vy, or Vz.

In various embodiments, ion energy is energy of ions formed withinplasma of a plasma chamber.

In some embodiments, when multiple RF generators are on, the Vpeak usedto calculate the wafer bias is that of the RF generator having thelowest frequency among all RF generators. For example, Vpeak is equal toVx. In various embodiments, when multiple RF generators are on, theVpeak used to calculate the wafer bias is that of the RF generatorhaving the highest frequency. For example, Vpeak is equal to Vz. Invarious embodiments, when multiple RF generators are on, the Vpeak usedto calculate the wafer bias is that of the RF generator having afrequency between the lowest frequency and the highest frequency. Forexample, Vpeak is equal to Vy. In several embodiments, Vpeak is a peakvoltage of a statistical value, e.g., median, mean, etc., of peak RFvoltages of the RF generators that are on. The ion energy calculated inthis manner removes a need to use an expensive voltage probe equipmentto measure the Vpeak and also removes a need to use a bias compensationcircuit to measure the wafer bias. The voltage probe used to measure theVpeak may be inaccurate. An example of the bias compensation circuitincludes a silicon carbide pin. The ion energy determined using variousembodiments of the present disclosure results in a low measured timebetween failures (MTBF).

It should be noted that in some embodiments, a value of the ion energyis stored in the storage HU 162.

It is further noted that although the above-described operations aredescribed with reference to a parallel plate plasma chamber, e.g., acapacitively coupled plasma chamber, etc., in some embodiments, theabove-described operations apply to other types of plasma chambers,e.g., a plasma chamber including an inductively coupled plasma (ICP)reactor, a transformer coupled plasma (TCP) reactor, conductor tools,dielectric tools, a plasma chamber including an electron-cyclotronresonance (ECR) reactor, etc. For example, the x MHz RF generator andthe y MHz RF generator are coupled to an inductor within the ICP plasmachamber.

It is also noted that although the operations above are described asbeing performed by the processor of the host system 130 (FIG. 1), insome embodiments, the operations may be performed by one or moreprocessors of the host system 130 or by multiple processors of multiplehost systems.

It should be noted that although the above-described embodiments relateto providing an RF signal to the lower electrode of the ESC 177 (FIGS. 1and 18) and to the lower electrode of the ESC 192 (FIG. 11), andgrounding the upper electrodes 179 and 264 (FIGS. 1 and 11), in severalembodiments, the RF signal is provided to the upper electrodes 179 and264 while the lower electrodes of the ESCs 177 and 192 are grounded.

Embodiments described herein may be practiced with various computersystem configurations including hand-held hardware units, microprocessorsystems, microprocessor-based or programmable consumer electronics,minicomputers, mainframe computers and the like. The embodiments canalso be practiced in distributed computing environments where tasks areperformed by remote processing hardware units that are linked through anetwork.

With the above embodiments in mind, it should be understood that theembodiments can employ various computer-implemented operations involvingdata stored in computer systems. These operations are those requiringphysical manipulation of physical quantities. Any of the operationsdescribed herein that form part of the embodiments are useful machineoperations. The embodiments also relates to a hardware unit or anapparatus for performing these operations. The apparatus may bespecially constructed for a special purpose computer. When defined as aspecial purpose computer, the computer can also perform otherprocessing, program execution or routines that are not part of thespecial purpose, while still being capable of operating for the specialpurpose. In some embodiments, the operations may be processed by ageneral purpose computer selectively activated or configured by one ormore computer programs stored in the computer memory, cache, or obtainedover a network. When data is obtained over a network, the data may beprocessed by other computers on the network, e.g., a cloud of computingresources.

One or more embodiments can also be fabricated as computer-readable codeon a non-transitory computer-readable medium. The non-transitorycomputer-readable medium is any data storage hardware unit that canstore data, which can be thereafter be read by a computer system.Examples of the non-transitory computer-readable medium include harddrives, network attached storage (NAS), ROM, RAM, compact disc-ROMs(CD-ROMs), CD-recordables (CD-Rs), CD-rewritables (CD-RWs), magnetictapes and other optical and non-optical data storage hardware units. Thenon-transitory computer-readable medium can include computer-readabletangible medium distributed over a network-coupled computer system sothat the computer-readable code is stored and executed in a distributedfashion.

Although the method operations in the flowchart of FIG. 2, FIG. 13, FIG.15, and FIG. 17 above were described in a specific order, it should beunderstood that other housekeeping operations may be performed inbetween operations, or operations may be adjusted so that they occur atslightly different times, or may be distributed in a system which allowsthe occurrence of the processing operations at various intervalsassociated with the processing, as long as the processing of the overlayoperations are performed in the desired way.

One or more features from any embodiment may be combined with one ormore features of any other embodiment without departing from the scopedescribed in various embodiments described in the present disclosure.

Although the foregoing embodiments have been described in some detailfor purposes of clarity of understanding, it will be apparent thatcertain changes and modifications can be practiced within the scope ofthe appended claims. Accordingly, the present embodiments are to beconsidered as illustrative and not restrictive, and the embodiments arenot to be limited to the details given herein, but may be modifiedwithin the scope and equivalents of the appended claims.

The invention claimed is:
 1. A method comprising: receiving, by a hostcomputer system, a first complex voltage and current measured at anoutput of a radio frequency (RF) generator, wherein the output of the RFgenerator is coupled via an RF cable to an input of an impedancematching circuit, wherein the impedance matching circuit is coupled to achuck of a plasma chamber via an RF transmission line; generating, bythe host computer system, an impedance matching model based on one ormore electrical components defined in the impedance matching circuit,the impedance matching model having an input and an output, the input ofthe impedance matching model receiving the first complex voltage andcurrent; generating, by the host computer system, an RF transmissionmodel based on one or more electrical components defined in the RFtransmission line, the RF transmission model having an input and anoutput, the input of the RF transmission model coupled to the output ofthe impedance matching model; generating, by the host computer system, achuck model based on one or more electrical components defined in thechuck, the chuck model having an input and an output, the input of thechuck model coupled to the output of the RF transmission model;propagating, by the host computer system, the first complex voltage andcurrent through the impedance matching model, the RF transmission model,and the chuck model to determine a second complex voltage and current atthe output of the chuck model; determining, by the host computer system,a wafer bias based on the second complex voltage and current; andcontrolling the RF generator based on the wafer bias.
 2. The method ofclaim 1, further comprising: determining, by the host computer system,an ion energy based on the wafer bias; and controlling the RF generatorbased on the ion energy.
 3. The method of claim 1, wherein the firstcomplex voltage and current is received by the host computer system froma voltage and current probe, wherein the voltage and current probe iscoupled to the output of the RF generator and to the host computersystem, wherein the voltage and current probe is located within the RFgenerator.
 4. The method of claim 1, wherein the impedance matchingmodel has a similar characteristic as that of the impedance matchingcircuit, wherein the RF transmission model has a similar characteristicas that of the RF transmission line, and the chuck model has a similarcharacteristic as that of the chuck.
 5. The method of claim 1, whereinthe impedance matching model includes one or more elements thatcorrespond to the one or more electrical components of the impedancematching circuit, wherein the one or more electrical components of theimpedance matching circuit includes a plurality of electricalcomponents, wherein the one or more elements of the impedance matchingmodel include a plurality of elements, wherein the plurality of elementsare coupled with each other in the same manner in which the plurality ofelectrical components are coupled with each other.
 6. The method ofclaim 1, wherein the RF transmission model includes one or more elementsthat correspond to the one or more electrical components of the RFtransmission line, wherein the one or more electrical components of theRF transmission model includes a plurality of electrical components,wherein the one or more elements of the RF transmission model include aplurality of elements, wherein the plurality of elements are coupledwith each other in the same manner in which the plurality of electricalcomponents are coupled with each other.
 7. The method of claim 1,wherein the chuck model includes one or more elements that correspond tothe one or more electrical components of the chuck, wherein the one ormore electrical components of the chuck model includes a plurality ofelectrical components, wherein the one or more elements of the chuckmodel include a plurality of elements, wherein the plurality of elementsare coupled with each other in the same manner in which the plurality ofelectrical components are coupled with each other.
 8. The method ofclaim 1, wherein said propagating comprises: calculating a firstdirectional sum of the first complex voltage and current and one or morevoltages and currents associated with one or more elements of theimpedance matching model between the input of the impedance matchingmodel and the output of the impedance matching model to generate a firstintermediate complex voltage and current at the output of the impedancematching model; calculating a second directional sum of the firstintermediate complex voltage and current and one or more voltages andcurrents associated with one or more elements between the input of theRF transmission model and the output of the RF transmission model togenerate a second intermediate complex voltage and current at the outputof the RF transmission model; and calculating a third directional sum ofthe second intermediate complex voltage and current and one or morevoltages and currents associated with one or more elements between theinput of the chuck model and the output of the chuck model to generatethe second complex voltage and current at the output of the chuck model.9. The method of claim 1, further comprising: determining, by the hostcomputer system, an ion energy based on the wafer bias and a peakvoltage identified from the second complex voltage and current.
 10. Acontroller comprising: a logic configured to: receive a first complexvoltage and current measured at an output of a radio frequency (RF)generator, wherein the output of the RF generator is coupled via an RFcable to an input of an impedance matching circuit, wherein theimpedance matching circuit is coupled to a chuck of a plasma chamber viaan RF transmission line; generate an impedance matching model based onone or more electrical components defined in the impedance matchingcircuit, the impedance matching model having an input and an output, theinput of the impedance matching model receiving the first complexvoltage and current; generate an RF transmission model based on one ormore electrical components defined in the RF transmission line, the RFtransmission model having an input and an output, the input of the RFtransmission model coupled to the output of the impedance matchingmodel; generate a chuck model based on one or more electrical componentsdefined in the chuck, the chuck model having an input and an output, theinput of the chuck model coupled to the output of the RF transmissionmodel; propagate the first complex voltage and current through theimpedance matching model, the RF transmission model, and the chuck modelto determine a second complex voltage and current at the output of thechuck model; determine a wafer bias based on the second complex voltageand current; and control the RF generator based on the wafer bias; and amemory device coupled to the logic for storing the impedance matchingmodel, the RF transmission model, and the chuck model.
 11. Thecontroller of claim 10, wherein the logic is configured to: determine anion energy based on the wafer bias; and control the RF generator basedon the ion energy.
 12. The controller of claim 10, wherein the firstcomplex voltage and current is received by the logic from a voltage andcurrent probe, wherein the voltage and current probe is coupled to theoutput of the RF generator and to the logic, wherein the voltage andcurrent probe is located within the RF generator.
 13. The controller ofclaim 10, wherein the impedance matching model has a similarcharacteristic as that of the impedance matching circuit, wherein the RFtransmission model has a similar characteristic as that of the RFtransmission line, and the chuck model has a similar characteristic asthat of the chuck.
 14. The controller of claim 10, wherein the impedancematching model includes one or more elements that correspond to the oneor more electrical components of the impedance matching circuit, whereinthe one or more electrical components of the impedance matching circuitincludes a plurality of electrical components, wherein the one or moreelements of the impedance matching model include a plurality ofelements, wherein the plurality of elements are coupled with each otherin the same manner in which the plurality of electrical components arecoupled with each other.
 15. The controller of claim 10, wherein the RFtransmission model includes one or more elements that correspond to theone or more electrical components of the RF transmission line, whereinthe one or more electrical components of the RF transmission modelincludes a plurality of electrical components, wherein the one or moreelements of the RF transmission model include a plurality of elements,wherein the plurality of elements are coupled with each other in thesame manner in which the plurality of electrical components are coupledwith each other.
 16. The controller of claim 10, wherein the chuck modelincludes one or more elements that correspond to the one or moreelectrical components of the chuck, wherein the one or more electricalcomponents of the chuck model includes a plurality of electricalcomponents, wherein the one or more elements of the chuck model includea plurality of elements, wherein the plurality of elements are coupledwith each other in the same manner in which the plurality of electricalcomponents are coupled with each other.
 17. The controller of claim 10,wherein to propagate, the logic is configured to: calculate a firstdirectional sum of the first complex voltage and current and one or morevoltages and currents associated with one or more elements of theimpedance matching model between the input of the impedance matchingmodel and the output of the impedance matching model to generate a firstintermediate complex voltage and current at the output of the impedancematching model; calculate a second directional sum of the firstintermediate complex voltage and current and one or more voltages andcurrents associated with one or more elements between the input of theRF transmission model and the output of the RF transmission model togenerate a second intermediate complex voltage and current at the outputof the RF transmission model; and calculate a third directional sum ofthe second intermediate complex voltage and current and one or morevoltages and currents associated with one or more elements between theinput of the chuck model and the output of the chuck model to generatethe second complex voltage and current at the output of the chuck model.18. The controller of claim 10, wherein the logic is configured todetermine an ion energy based on the wafer bias and a peak voltageidentified from the second complex voltage and current.
 19. A plasmasystem comprising: a radio frequency (RF) generator configured togenerate an RF signal; an impedance matching circuit coupled to the RFgenerator via an RF cable, wherein the RF cable is coupled to an outputof the RF generator and an input of the impedance matching circuit; aplasma chamber coupled to the impedance matching circuit via an RFtransmission line; a host computer system coupled to the RF generator,wherein the host computer system is configured to: receive a firstcomplex voltage and current measured at the output of the RF generator;generate an impedance matching model based on one or more electricalcomponents defined in the impedance matching circuit, the impedancematching model having an input and an output, the input of the impedancematching model receiving the first complex voltage and current; generatean RF transmission model based on one or more electrical componentsdefined in the RF transmission line, the RF transmission model having aninput and an output, the input of the RF transmission model coupled tothe output of the impedance matching model; generate a chuck model basedon one or more electrical components defined in the chuck, the chuckmodel having an input and an output, the input of the chuck modelcoupled to the output of the RF transmission model; propagate the firstcomplex voltage and current through the impedance matching model, the RFtransmission model, and the chuck model to determine a second complexvoltage and current at the output of the chuck model; determine a waferbias based on the second complex voltage and current; and control the RFgenerator based on the wafer bias.
 20. The plasma system of claim 19,wherein the host computer system is configured to: determine an ionenergy based on the wafer bias; and control the RF generator based onthe ion energy.
 21. The plasma system of claim 19, wherein the firstcomplex voltage and current is received by the host computer system froma voltage and current probe, wherein the voltage and current probe iscoupled to the output of the RF generator and to the host computersystem, wherein the voltage and current probe is located within the RFgenerator.
 22. The plasma system of claim 19, wherein the impedancematching model has a similar characteristic as that of the impedancematching circuit, wherein the RF transmission model has a similarcharacteristic as that of the RF transmission line, and the chuck modelhas a similar characteristic as that of the chuck.
 23. The plasma systemof claim 19, wherein the impedance matching model includes one or moreelements that correspond to the one or more electrical components of theimpedance matching circuit, wherein the one or more electricalcomponents of the impedance matching circuit includes a plurality ofelectrical components, wherein the one or more elements of the impedancematching model include a plurality of elements, wherein the plurality ofelements are coupled with each other in the same manner in which theplurality of electrical components are coupled with each other.
 24. Theplasma system of claim 19, wherein the RF transmission model includesone or more elements that correspond to the one or more electricalcomponents of the RF transmission line, wherein the one or moreelectrical components of the RF transmission model includes a pluralityof electrical components, wherein the one or more elements of the RFtransmission model include a plurality of elements, wherein theplurality of elements are coupled with each other in the same manner inwhich the plurality of electrical components are coupled with eachother.
 25. The plasma system of claim 19, wherein the chuck modelincludes one or more elements that correspond to the one or moreelectrical components of the chuck, wherein the one or more electricalcomponents of the chuck model includes a plurality of electricalcomponents, wherein the one or more elements of the chuck model includea plurality of elements, wherein the plurality of elements are coupledwith each other in the same manner in which the plurality of electricalcomponents are coupled with each other.
 26. The plasma system of claim19, wherein to propagate, the host computer system is configured to:calculate a first directional sum of the first complex voltage andcurrent and one or more voltages and currents associated with one ormore elements of the impedance matching model between the input of theimpedance matching model and the output of the impedance matching modelto generate a first intermediate complex voltage and current at theoutput of the impedance matching model; calculate a second directionalsum of the first intermediate complex voltage and current and one ormore voltages and currents associated with one or more elements betweenthe input of the RF transmission model and the output of the RFtransmission model to generate a second intermediate complex voltage andcurrent at the output of the RF transmission model; and calculate athird directional sum of the second intermediate complex voltage andcurrent and one or more voltages and currents associated with one ormore elements between the input of the chuck model and the output of thechuck model to generate the second complex voltage and current at theoutput of the chuck model.
 27. The plasma system of claim 19, whereinthe host computer system is configured to determine an ion energy basedon the wafer bias and a peak voltage identified from the second complexvoltage and current.